Finitary groups and Krull dimension over the integers
Let M be any Abelian group. We make a detailed study for reasons explained in the Introduction of the normal subgroup F∞AutM={g∈AutM:M(g−1)isaminimaxgroup} of the automorphism group AutM of M. The conclusions, although slightly weaker than one would hope, in that they do not fully explain the comm...
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Wehrfritz, B.A. 2020-02-13T12:28:32Z 2020-02-13T12:28:32Z 2006 Finitary groups and Krull dimension over the integers / B.A. Wehrfritz // Український математичний журнал. — 2006. — Т. 58, № 10. — С. 1310–1325. — Бібліогр.: 11 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165413 512.5 Let M be any Abelian group. We make a detailed study for reasons explained in the Introduction of the normal subgroup F∞AutM={g∈AutM:M(g−1)isaminimaxgroup} of the automorphism group AutM of M. The conclusions, although slightly weaker than one would hope, in that they do not fully explain the common behavior of the finitary and the Artinian-finitary subgroups of AutM, are certainly stronger than one might reasonably expect. Our main focus is on residual properties and unipotence. Нехай M — будь-яка абелева група. З мотивів, що вказані у вступі до статті, детально вивчено нормальну підгрупу F∞Aut M={g∈AutM:M(g−1)мінімаксна група} групи Aut M автоморфізмів M. Хоча отримані результати слабкіші, ніж хотілося б, оскільки вони не повністю пояснюють загальну поведінку фінітарних та артіново-фінітарних підгруп групи Aut M, але вони беззаперечно сильніші, ніж можна було б очікувати. Головну увагу приділено залишковим властивостям та уніпотентності. en Інститут математики НАН України Український математичний журнал Статті Finitary groups and Krull dimension over the integers Фінітарні групи i розмірність Крулля над цілими числами Article published earlier |
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Finitary groups and Krull dimension over the integers |
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Finitary groups and Krull dimension over the integers Wehrfritz, B.A. Статті |
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Finitary groups and Krull dimension over the integers |
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Finitary groups and Krull dimension over the integers |
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Finitary groups and Krull dimension over the integers |
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Finitary groups and Krull dimension over the integers |
| title_sort |
finitary groups and krull dimension over the integers |
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Wehrfritz, B.A. |
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Wehrfritz, B.A. |
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Фінітарні групи i розмірність Крулля над цілими числами |
| description |
Let M be any Abelian group. We make a detailed study for reasons explained in the Introduction of the normal subgroup
F∞AutM={g∈AutM:M(g−1)isaminimaxgroup}
of the automorphism group AutM of M. The conclusions, although slightly weaker than one would hope, in that they do not fully explain the common behavior of the finitary and the Artinian-finitary subgroups of AutM, are certainly stronger than one might reasonably expect. Our main focus is on residual properties and unipotence.
Нехай M — будь-яка абелева група. З мотивів, що вказані у вступі до статті, детально вивчено нормальну підгрупу
F∞Aut M={g∈AutM:M(g−1)мінімаксна група}
групи Aut M автоморфізмів M. Хоча отримані результати слабкіші, ніж хотілося б, оскільки вони не повністю пояснюють загальну поведінку фінітарних та артіново-фінітарних підгруп групи Aut M, але вони беззаперечно сильніші, ніж можна було б очікувати. Головну увагу приділено залишковим властивостям та уніпотентності.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/165413 |
| citation_txt |
Finitary groups and Krull dimension over the integers / B.A. Wehrfritz // Український математичний журнал. — 2006. — Т. 58, № 10. — С. 1310–1325. — Бібліогр.: 11 назв. — англ. |
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AT wehrfritzba finitarygroupsandkrulldimensionovertheintegers AT wehrfritzba fínítarnígrupiirozmírnístʹkrullânadcílimičislami |
| first_indexed |
2025-11-25T21:29:29Z |
| last_indexed |
2025-11-25T21:29:29Z |
| _version_ |
1850557887644958720 |
| fulltext |
UDC 512.5
B. A. F. Wehrfritz (School Math. Sci. Queen Mary, Univ. London, England)
FINITARY GROUPS AND KRULL DIMENSION
OVER THE INTEGERS
FINITARNI HRUPY I ROZMIRNIST| KRULLQ
NAD CILYMY ÇYSLAMY
Let M be any abelian group. We make a detailed study for reasons explained in the Introduction of the
normal subgroup
F∞ Aut M = { g ∈ Aut M : M( g – 1 ) is a minimax group }
of the automorphism group Aut M of M. The conclusions, although slightly weaker than one would
hope, in that they do not fully explain the common behaviour of the finitary and the Artinian-finitary
subgroups of Aut M, are certainly stronger that one might reasonably expect. Our main focus is on
residual properties and unipotence.
Nexaj M — bud\-qka abeleva hrupa. Z motyviv, wo vkazani u vstupi do statti, detal\no vyvçeno
normal\nu pidhrupu
F∞ Aut M = { g ∈ Aut M : M( g – 1 ) — minimaksna hrupa}
hrupy Aut M avtomorfizmiv M. Xoça otrymani rezul\taty slabkißi, niΩ xotilosq b, oskil\ky
vony ne povnistg poqsnggt\ zahal\nu povedinku finitarnyx ta artinovo-finitarnyx pidhrup
hrupy Aut M, ale vony bezzapereçno syl\nißi, niΩ moΩna bulo b oçikuvaty. Holovnu uvahu pry-
dileno zalyßkovym vlastyvostqm ta unipotentnosti.
1. Introduction. For M an abelian group we study in this paper the normal subgroup
F∞ Aut M = { g ∈ Aut M : M ( g – 1 ) has Krull dimension over the integers}
of the automorphism group Aut M of M. On Page 255 of Vol. 2 of this book [1]
Fuchs appeals for a greater understanding of the automorphism group of an abelian
group and in particular of the normal subgroups of such an automorphism group. This
paper could be interpreted as a small contribution to this appeal.
However, our motivation has come from more general ring and module theoretic
considerations. If M is any module over the commutative ring R, set
F AutR M = { g ∈ AutR M : M ( g – 1 ) is R-Noetherian},
F1 AutR M = { g ∈ AutR M : M ( g – 1 ) is R-Artinian}
and
F∞ AutR M = { g ∈ AutR M : M ( g – 1 ) has Krull dimension as R-module}.
These are three examples of an infinite number of (generalized) finitary groups, the
first being the finitary group and the second the Artinian-finitary group. They are
almost the only finitary groups relevant to this paper, but for accounts of the general
set-up, see [2] and [3], Section 2.
Now F AutR M and F1 Aut RM are both subgroups of F∞ Aut RM. Also they have
very many properties in common, see [4 – 8] and especially [9]. Now do these
properties come from a common factor? Specifically, do they have these properties in
common because F∞ Aut RM also enjoys these properties? We propose to study this
question here just in the special case where R is the ring Z of the integers. The
answer it turns out in this case is frequently no, but then F∞ Aut M only just avoids
satisfying them. While working over the integers we suppress the suffix Z and just
© B. A. F. WEHRFRITZ, 2006
1310 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
FINITARY GROUPS AND KRULL DIMENSION OVER THE INTEGERS 1311
write F Aut M, F1 Aut M and F∞ Aut M. The first two of these three groups we have
already investigated in some detail in [5]. Thus this paper is devoted the third.
Notationally it is convenient in this Introduction to include the possibility of an
arbitrary commutative ring R . Thus just throughout this section R denotes a
commutative ring and M an R-module. Mostly we only consider the additive
structure, that is the Z-module structure, of M and hence we mainly refer to F Aut M
rather than F AutR M, and similarly with the other finitary groups.
Suppose M has Krull dimension over Z. Then every section of M has finite
uniform dimension and so M has a free abelian subgroup of finite rank modulo which
M is a direct sum of a finite number of finite cyclic groups and Prüfer groups. Such a
group is usually called a minimax group. Conversely, a minimax group is Noetherian
by Artinian and has Krull dimension (at most one). Thus we propose to call F∞ Aut M
the minimax-finitary group of M (over Z ).
We will see below, and it is very easy, that there is a natural exact sequence
1 → F1 Aut M → F∞ Aut M → F G L ( Q ⊗ M ).
Of course Q denotes the rationals. If V is a vector space over some field F, then
F G L ( V ) denotes the finitary general linear group over V . Thus F∞ Aut M is
Artinian-finitary by finitary-linear. The proof also yields that if T denotes the Z-
torsion submodule of M , as it will throughout this paper, then A = CG ( M / T ) ∩
∩ CG ( T ) is an abelian normal subgroup of G such that G A/ embeds into
F G L ( Q ⊗ M ) × F1 Aut T. These simple remarks produce a crude reduction of the
study of F∞ Aut M to that of finitary-linear and Artinian-finitary groups. For example,
it is immediate that periodic subgroups of F∞ Aut M are locally finite. But it is a quite
crude reduction; for example, it only produces a number of metabelian subgroups of
F∞ Aut M, while a more careful study shows that there exist nilpotent-of-class-at-most-
2 (occasionally even abelian) subgroups that fulfill the same roles.
We now summarise our main results. We start with a consideration of residual
properties. Both F Aut M and F 1 Aut M are locally residually finite and also are
abelian by residually locally finite ([5], 10
1, 101
, 11
1 and 111
). Further both F Aut M
and F1 Aut M are, modulo a central unipotent subgroup, residually (finitary over
commutative Noetherian rings), see [9], Proposition 8.11. None of these hold for
F∞ Aut M as we show with examples. However, they nearly hold.
Theorem 1. Let M be an abelian group.
a) F∞ Aut M is abelian by locally residually finite and hence locally (abelian by
residually finite). If G is any finitely generated subgroup of F∞ Aut M, then G
has normal subgroups A ≤ H such that A is abelian, H A/ is residually a finite
nilpotent π-group for some finite set π of primes and G / H is finite.
b) F∞ Aut M is nilpotent-of-class-at-most-2 by residually locally finite.
c) F∞ Aut M is unipotent-abelian by residually ( finitary over the rationals Q
and finite images of Z ).
As we have stated the theorem, Part a) it easy. However with a little more work we
can be much more precise about the set π of primes, see the proof of 3.2. We show
with examples that the group G of Part a) of Theorem 1 need not be central by
residually finite (let alone residually finite). Also in Part b) the periodic residual of
G ≤ F∞ Aut M, although nilpotent of class at most 2, need not be abelian, or central (in
G ) by abelian or abelian by central (in G ). In Part c) the unipotent abelian normal
subgroup cannot be replaced by a central subgroup of G.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1312 B. A. F. WEHRFRITZ
Any subgroup G of F∞ Aut M has a unipotent radical, which we denote by u ( G ).
Precisely what we mean by this is spelled out by the next theorem. For the latter parts
of this theorem we need some further notation. By ψ we denote the natural map of
F∞ Aut M into F G L ( Q ⊗ M ) discussed above. By an s-subgroup of Aut M (or of
AutR M ) we mean a subgroup each of whose finitely generated subgroups stabilizes an
ascending series of submodules of M. Unlike the unipotent radical, the s-radical s ( G )
of G exists under very general conditions and often has to be used as a replacement
for a unipotent radical, see [2], especially Section 4. Finally 5.2 of [8] introduces a
collection of maps ψλ attached to an R-module M and a subgroup G of F1 AutR M.
Since we shall only user these maps once and that only in order to apply [8], 5.2, we
will not burden the reader with their precise construction. See [8] for details.
Theorem 2. Let G be a subgroup of F∞ Aut M. Then G has a unipotent
radical. Specifically G has a normal subgroup u ( G ) with the following
properties.
a) u ( G ) is a unipotent normal subgroup of G containing every unipotent
normal subgroup of G.
b) G / u ( G ) is a subdirect product of irreducible finitary linear groups over Q ,
G F ( p ) and Q p , the p-adic numbers, for various primes p.
c) Let H be a finitely G-generated normal subgroup of G. Then u ( H ) =
= H ∩ u ( G ) and u ( H ) is nilpotent. In particular u ( G ) lies in the Fitting
subgroup of G and u ( G ) is a Fitting group.
d) If H is an ascendent subgroup of G, then u ( H ) is an ascendent subgroup
of u ( G ).
e) u ( G ) ≤ s ( G ) and s ( G ) ψ ≤ u ( G ψ ).
f) u ( G ) = G ∩ ( u ( G ψ ) ) ψ
–
1 ∩ ( 1 + ∩ λ ker ψλ ), where the ψλ are the maps of
[8], 5.2, with respect to the Z-torsion submodule T of M and the subgroup
G | T of Aut T. (The map ψ is described above.)
We can now state a further residual result. It effectively copies the corresponding
results for F AutR M and F1 AutR M, which we have previously only written out for
local rings (see [9], Theorem 5), since for arbitrary commutative rings R it becomes
very cumbersome. The pattern, however, is clearly the same. Note that B remains
abelian; it does not get replaced by a subgroup that is nilpotent of class at most 2 or
even one that is metabelian, as Theorem 1 might suggest. The analogues of Parts c)
and g) of [9] (Theorem 5) we have incorporated into Theorem 2 above and hence have
not repeated below. Part f) of Theorem 3 below has no analogue in [9], Theorem 5,
although it does in [9], Theorem 6a).
Theorem 3. Let G be a subgroup of F∞ Aut M. Then G has a normal series
〈 1 〉 ≤ B ≤ K ≤ u ( G ) ≤ G, where
a) B is abelian;
b) K / B is residually (nilpotent of finite exponent);
c) u ( G ) / B is a subdirect product of a torsion-free group and a residually
periodic group; it is also abelian by residually periodic;
d) u ( G ) / K is a subdirect product of unipotent finitary linear groups over Q ,
G F ( p ) and Q p for various primes p;
e) G / K is a subdirect product of finitary linear groups over Q , G F ( p ) and
Q p for various primes p;
f) K stabilizes an ascending series in M.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
FINITARY GROUPS AND KRULL DIMENSION OVER THE INTEGERS 1313
Information about the sections G / u ( G ) and u ( G ) of G omitted from Theorem
3 is given in Theorem 2. Our final theorem concerns the structure of unipotent groups
and unipotent radicals. To state it reasonably succinctly we use the following notation
for group classes and operators. A denotes the class of abelian groups, N2 the class
of nilpotent groups of class at most 2, N the class of nilpotent groups, F the class of
finite groups and E the class of groups of finite exponent. Also if X and Y are any
group classes, then S ( X × Y ) denotes the class of all subgroups of direct products of
an X-group and a Y-group and L X (resp. Ln X ) denotes the class of all groups G
each of whose finite subsets lies in an X-subgroup (resp. normal X-subgroup) of G.
Theorem 4. Let G be a subgroup of F∞ Aut M.
a) If G is unipotent, then G is periodic-abelian by torsion-free by periodic.
b) Let X be a finite subset of G and U the maximal periodic subgroup of the
unipotent radical of the normal subgroup 〈 X
G
〉 of G generated by X. Then
U ∈ N ∩ N2
E ∩ A. S ( A × E ) ≤ A E A.
c) If G is periodic and unipotent, then
G ∈ Ln ( N ∩ N2
E ∩ A E A ) and G′ ∈ Ln ( N ∩ A E ).
Since in Part a) the group G is locally nilpotent and locally nilpotent groups are
always periodic by torsion-free, Part a) is saying that G is not far from being a
subgroup group of a direct product of a periodic group and a torsion-free group. Since
we already know by [9], Theorem 2, that in Part a) above the conclusions hold
whenever G is contained in either F Aut M or F1 Aut M, this is an instance where an
abelian normal subgroup need not be replaced in the F∞ Aut M case by, say, a
nilpotent normal subgroup of class at most 2.
In Part b) if G lies in either F Aut M or F1 Aut M then
U ∈ N ∩ A
E ∩ E A ∩ ( A ∩ E ). S ( A × E )
by [9], Theorem 4, so again we get a weakening in Part b) of our Theorem 4, but
perhaps its from is not so predictable. Similar remarks apply to a comparison between
Part c) of Theorem 4 with the Corollary to Theorem 4 of [9].
Theorem 2 is proved in Section 2 below, Theorems 1 and 3 in Section 3 and
Theorem 4 in Section 4. In our final section, Section 5, we consider a variant of the
above, namely that arising from considering extensions of Artinian groups by
Noetherian groups rather than the reverse. This section also contains some further
counter examples.
2. General results.
2.1. Lemma. Let G = F∞ Aut M.
a) There is a natural exact sequence 1 → F1 Aut M → F∞ Aut M → F G L ( Q ⊗
⊗ M ).
b) If A = CG ( M / T ) ∩ CG ( T ), then A is abelian and G / A embeds into the
direct product F G L ( Q ⊗ M ) × F1 Aut T. ( T recall is the Z-torsion submodule of
M.) In particular G / A is locally quasilinear.
c) F1 Aut M embeds into F AutJ N, where J =
p p∏ Z is the profinite closure
of Z and N is some J-module.
d) Periodic subgroups of F∞ Aut M are locally finite.
So by a) and c) the group F∞ Aut M is an extension of a finitary group over J by a
finitary group over Q .
Proof. a) Let L = F∞ Aut M and g ∈ G. If g ∈ F1 Aut M, then M ( g – 1 ) is
torsion and g ∈ CG ( M / T ). Conversely, if g ∈ CG ( M / T ), then M ( g – 1 ) is
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1314 B. A. F. WEHRFRITZ
minimax, torsion and hence Artinian. Thus g ∈ F1 Aut M and so F1 Aut M =
= CG ( M / T ). Clearly G / CG ( M / T ) embeds naturally into F∞ Aut ( M / T ) and
M / T embeds in the obvious way into Q ⊗ ( M / T ) � Q ⊗ M . This leads to an
embedding of F∞ Aut ( M / T ) into F∞ Aut Q ( Q ⊗ M ) = F G L ( Q ⊗ M ). Part a)
follows.
b) G / CG ( M / T ) embeds into F G L ( Q ⊗ M ) as in Part a) and G / CG ( T )
embeds into F∞ Aut T = F1 Aut T, again see Part a). By stability theory A is abelian.
By N4 and A3 of [7] the group G / A is locally quasilinear. Part b) follows.
c) A application of the proof of A1 of [7] to our M yields Part c).
d) This follows from a), the corresponding results for finitary and Artinian-finitary
groups ([7], N4 and A3) and the simple and well-known fact that extensions of locally
finite groups by locally finite groups are locally finite.
2.2. Proof of Theorem 2. a) & f). Define u ( G ) by the formula in f ). Certainly
u ( G ) is normal in G and if U is a unipotent normal subgroup of G, then Uψ ≤
≤ u ( G ψ ) and each U ψ λ = 〈 1 〉. Thus U ≤ u ( G ).
Let g ∈ u ( G ). Then g ψ is unipotent, so ( g ψ – 1 )
m = 0 for some m. It follows
from the definition of ψ that M ( g – 1 )
m ≤ T. Also g acts unipotently on T by 5.2
of [8], say T ( g – 1 )
n = { 0 }. Trivially then ( g – 1 )
m
+
n = 0 and so u ( G ) is unipotent.
b) It follows from [8], 5.2, and the finitary linear case that G / u ( G ) is a subdirect
product of certain irreducible finitary linear images of G. We have only to specify the
ground fields. From G ψ only the rationals Q arise. One has to check through the
proofs in [8] to see that the only ground fields that arise from the ψ λ are the G F ( p )
and Q p for various primes p.
c) The group u ( H ) is normal in G, so u ( H ) ≤ H ∩ u ( G ) and H ∩ u ( G ) is
unipotent and normal in H. Therefore u ( H ) = H ∩ u ( G ). If u denotes the
augmentation ideal of u ( H ), then T u
s = { 0 } for some s by [8], 5.2, and M u
t ≤ T
for some t by [10], 3.7 (or the finitary linear version if you prefer). Thus M u
s
+
t =
= { 0 } and all of Part c) is a consequence of this.
d) The usual transfinite induction yields this (cf. [4], 3.8, or [2], 4.10).
e) By the proof of c), if U is a finitely generated subgroup of u ( G ), then U
stabilizes a finite series in M. Hence u ( G ) is an s-subgroup, so u ( G ) ≤ s ( G ). By
4.2 and 4.3 of [2] we have that s ( G ) ψ is a s-subgroup and hence a unipotent
subgroup of F G L ( Q ⊗ M ). (Note that all nontrivial torsion-free abelian groups are
tight.) Thus s ( G ) ψ ≤ u ( G ψ ).
2.3. Remark. If M is minimax, then Aut M can contain a unipotent Prüfer
subgroup, indeed even a noncentral one. (If M is Noetherian or Artinian, then Aut M
contains no Prüfer subgroups.) For let M = Z ⊕ P, where P is a Prüfer p-group for
some prime p. Then P is isomorphic to the unipotent subgroup Q =
1
0 1
P
of
Aut M. If U is the group of units of the p-adic integers, then Q is a noncentral
Prüfer subgroup of the subgroup G = diag ( 1, U ). Q of Aut M. Further G is not
residually finite or even central by residually finite. This example is pertinent to both
2.4 and 2.5 below.
2.4. Lemma. Let M be an abelian minimax group and let spec M denote the
(finite) set of primes p such that M has a Prüfer p-group section. Set G =
= Aut M.
a) If p ∉ spec M, then i
ip M∩ is torsion.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
FINITARY GROUPS AND KRULL DIMENSION OVER THE INTEGERS 1315
b) If M is torsion-free then G is residually finite; in fact if p ∉ spec M, then
G is (residually finite-p) by finite.
c) If M is a p-group, then G is residually finite and again even (residually
finite-p) by finite.
d) Let π be any set of primes properly containing spec M. Then G has
normal subgroups A ≤ H with A abelian, H / A residually a finite nilpotent π-
group and G / H finite: G / A is residually finite.
Proof. a) For there is a free abelian subgroup F of M of finite rank with M / F
Artinian. The p-primary component of M / F is finite, of finite exponent p
e say. Let
x ∈ M with p
i + e x ∈ F. Then p
e x ∈ F and p
i + e x ∈ p
i F. Thus p
i + eM ∩ F ≤ p
i F and
consequently
i
ip M∩
∩ F ≤
i
ip F∩ = {0}.
Therefore i
ip M∩ embeds into the torsion group M / F and the claim follows.
b) By Part a) here
i
ip M∩ = {0}. Also each M / p
i M is finite. Set H =
= CG ( M / p M ). Then H is normal subgroup of G of finite index; in fact G / H
embeds into G L ( n, p ) for p
n = | M / p M |. Further [ p
i M, H ] = p
i [ M, H ] ≤ p
i + 1 M
for all i ≥ 0. Thus H / CH ( M / p
i M ) is a finite p-group. Part b) follows.
c) If H = C x M pxG ∈ ={ }( ): 0 , then H is a normal subgroup of G of finite
index and H is a residually a finite p-group; for H stabilizes the ascending series
x M p xi
i
∈ ={ } ≥
: 0
0
.
d) Let T denote the torsion subgroup of M and Tp the p-primary component of
T for the prime p. If Tp is finite, then G / CG ( Tp ) is finite. If Tp is infinite, then
p ∈ spec M and G / CG ( Tp ) is (residually finite-p ) by finite by Part c). By
hypothesis there is some prime p in π \ spec M. By Part b) the group G / CG ( M / T )
is (residually finite-p ) by finite. Set
A = CG ( M / T ) ∩
p
G pC T∩ ( ) = CG ( M / T ) ∩ CG ( T ).
Then A is abelian and G / A is (residually a finite nilpotent π-group) by finite.
Clearly G / A is residually finite.
2.5. Lemma. Let U denote the subgroup of Aut M generated by all its
unipotent elements of finite order and let C denote the subgroup of F∞ Aut M
generated by all its Prüfer subgroups. Set R = n
n M∩ and let D be the divisible
component of the torsion subgroup T of M.
a) [ M, U ] ≤ T and [ R, U ] = 〈 1 〉.
b) C ≤ U and [ T, C ] = { 0 }, while [ M, C ] ≤ D ≤ R.
c) C lies in the centre of U and hence is divisible.
d) U centralizes [ M, C ] and C centralizes [ M, U ].
Proof. a) This follows from [11], 2.2a) and 2.4a).
b) If P is Prüfer group, then any irreducible finitary linear image of P is finite
dimensional. Now GL ( n, Q ), G L ( n, Qp ) and G L ( n, p ) contain no Prüfer
subgroups, the first two by [8], 9.33, and the third trivially. Then by Theorem 2b) all
Prüfer subgroups of F∞ Aut M are unipotent. Consequently C ≤ U. That [ T, C ] =
= { 0 } and [ M, C ] ≤ D follow from [5], 13.
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1316 B. A. F. WEHRFRITZ
c) Now C embeds by b) into Hom ( M / T, D ) via c ∈ C mapping to m + T �
� m ( c – 1 ), and this embedding is a U-embedding. By Part a) this Hom group is
centralized by U. Therefore C lies in the centre of U; clearly then, C is divisible.
d) U centralizes R ≥ D ≥ [ M, C ] and C centralizes T ≥ [ M, U ] by Parts a)
and b).
2.6. Lemma. Let G be any subgroup of F∞ Aut M and let R and D be as
in 2.5.
a) If D = { 0 }, then M / CM ( G ) is residually finite.
b) D ≤ R and [ R, G ] ≤ D.
Proof. a) Let g ∈ G. Then M ( g – 1 ) is minimax. Since D = { 0 }, the torsion
subgroup of M ( g – 1 ) is finite, so it splits off from M ( g – 1 ), e.g. [1], Vol. 1, 27.5.
Consequently M ( g – 1 ) and hence M / CM ( g ) are residually finite (by 2.4a)). It
follows that M / CM ( G ) is residually finite.
b) Clearly D ≤ R , so R / D = n
n M D∩ /( ) . By Part a) we have R / D ≤
≤ C GM D/ ( ). That is, [ R, G ] ≤ D.
3. Residual properties.
3.1. Lemma. Let G be a finitely generated subgroup of F∞ Aut M with N =
= [ M, G ] torsion. Let π = π ( N ) = { primes p : N has an element of order p }.
Then G is (residually finite nilpotent g-group) by finite.
Proof. Here N is Artinian (so in particular π is finite). Then N = ⊕ ∈p pNπ ,
where Np is p-primary. It suffices to prove the result for G acting on each
M Nq p q/⊕ ≠ . Thus we may assume that N is a p-group for some prime p ,
necessarily of finite rank, r say. Let Si i{ } ≥0 be the upper socle series of N, so Si =
= Ann N ( p
i
) and
i iS∪ = N.
Let x ∈ M. Since G is finitely generated, so [ Z x, G ] ≤ Sk for some integer k
and then Z x G ≤ Z x + Sk . Hence additively Z x G is a cyclic extension of a finite p-
group of rank at most r. Suppose x has infinite order. Then
i
ip xG∩ Z = { 0 }, the
group
CG ( Z x G / p Z x G ) / CG ( Z x G )
is residually a finite p-group and ( G : CG ( Z x G / p Z x G ) ) ≤ l = GL r p( , )+ 1 . If x
is a p-element, the same conclusions hold (even with CG ( Z x G / p Z x G ) / CG ( Z x G ) a
finite p-group). Now suppose x is a p ′-element. Then x ( g – 1 ) is both a p-element
and a p ′-element for every g in G and so G centralizes x. This trivially the same
conclusions also hold in this case.
Let H be the intersection of all the normal subgroups of G of index at most l.
Since G is finitely generated, so ( G : H ) is finite. We claim that H is residually a
finite p-group. Let h ∈ H \ 〈1〉. There exists x in M with x h ≠ x and we can choose
such an x either of infinite order or a p-element or a p ′-element. Then H / CH ( Z x G )
is residually a finite p-group. Hence there is a normal subgroup L of H with H / L
a finite p-group and with h ∉ L. The result follows. (We can always choose such an
L normal in G if we wish, but actually it you chase through the above proof the L
you arrive at is automatically normal in G.)
3.2. Lemma. Let G be a finitely generated subgroup of F∞ Aut M. Then G
is abelian by (residually a finite nilpotent π-group) by finite, for some finite set π
of primes (depending on G ).
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FINITARY GROUPS AND KRULL DIMENSION OVER THE INTEGERS 1317
Proof. Set C = CM ( G ). By 2.1b) of [2] the group M / C is minimax. Let X / C
be the torsion submodule of M / C. By 2.4b) the group G / CG ( M / X ) is (residually
finite-nilpotent) by finite.
Now [ X, G ] is torsion; for if x ∈ X and g ∈ G there is a positive integer m with
m x ∈ C . Then m ( x ( g – 1 ) ) = ( m x ) ( g – 1 ) = 0, so x ( g – 1 ) is torsion and
consequently [ X, G ] is too. It follows from 3.1 that G / CG ( X ) is (residually finite-
nilpotent) by finite. Further A = CG ( M / X ) ∩ CG ( X ) is abelian. The existence of π
follows from 2.4b) and 3.1; in fact, we can choose π to be { p } ∪ π ( [ X, G ] ) for any
prime p not in spec ( M / X ). Note that
spec ( M / X ) ⊆ spec ( M / C ) = spec [ M, G ] ⊆ π ( [ M, G ] ).
3.3. Examples. In 6.3 of [6] we construct a minimax group M and a 3-generator
soluble subgroup G of Aut M = F∞ Aut M whose centre is a Prüfer 2-group. Clearly
G is not residually finite. The wreath product W of G be a cyclic group of order 2
embeds into Aut ( M ⊕ M ) and is not central by residually finite. Of course W is
finitely generated and soluble.
3.4. Lemma. Let G be subgroup of F∞ Aut M.
a) If M is torsion or residually of finite exponent, the G is residually (finitary
over finite images of Z ).
b) If M is torsion-free, then G is abelian by residually (finitary over finite
images of Z ).
c) In general G is (nilpotent of class at most 2) by residually (finitary over
finite images of Z ).
Proof. a) Here either M = n
x M nx≥ ∈ ={ }
1
0∪ : or { 0 } = n
nM≥1∩ . Clearly
either
n
GC x M nx
≥
∈ =
1
0∩ ( : ) = { 0 } or
n
GC M nM
≥
/( )
1
∩ = { 0 }.
Part a) follows.
b) Let R = n
nM≥1∩ . Then G / CG ( M / R ) is residually (finitary over finite
images of Z ) by a). Also [ R, G ] = { 0 } by 2.6b). Therefore CG ( M / R ) here is
abelian.
c) Set S / T =
n
n M T≥ /( )
1∩ . By 2.6b) again [ S, G ] ≤ T. Also G / CG ( M / S )
and G / CG ( T ) are residually (finitary over finite images of Z ) by Part a) and hence so
is G / N for N = CG ( M / S ) ∩ CG ( T ). Finally N stabilizes the series { 0 } ≤ T ≤ S ≤
≤ M, so N is nilpotent of class at most 2.
3.5. Examples. Let A = 〈 a1 , a2 〉 be a free abelian group of rank 2, let B be the
additive group of the rationals and let C be a Prüfer p-group for some prime p. Put
M = A ⊕ B ⊕ C. Set X = Hom ( A, B ) � B
(
2
)
, Y = Hom ( B, C ) and Z = Hom ( A,
C ) � C
(
2
)
. Note that Y is a copy of the additive group of Qp in a natural way. ( Y is
torsion-free and divisible, see G ) on p. 182 of Vol. 1 of [1], and is a Zp-module via the
action on C and hence is naturally a Qp-module; finally the exact sequence
0 → Z → Q → Q / Z → 0
yields exact sequences
0 → End C → Y → C → 0
and
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1318 B. A. F. WEHRFRITZ
0 → Zp → Y → C → 0,
so Y � Qp as Qp-module.)
Let u =
1 1
0 1
∈ G L ( 2, Z ), regarded as an automorphism of A via the given
basis a1 , a2 and let v be a p-adic unit (and hence an automorphism of C ) of
infinite order. Set
G =
u X Z
Y0 1
0 0 v
,
regarded as a subgroup of Aut M in the obvious way. Let ξ ∈ X, η ∈ Y and ζ ∈ Z.
Then M ( u – 1 ) = 〈 a2 〉 is cyclic, M ξ = 〈 a1 ξ, a2 ξ 〉 ≤ B again is cyclic, M η ≤ C is
Artinian, M ζ = 〈 a1 ζ, a2 ζ 〉 is finite cyclic and M ( v – 1 ) ≤ C is Artinian. Therefore
G ≤ F∞ Aut M.
Let H be a normal subgroup of G with G / H periodic. Then u
i and v
j lie in
H for some positive integers i and j. Thus H contains 〈 u
i G
〉 and hence contains
[ 1 + X, u
i
]. Define ξ1 ∈ X by a1 ξ1 = a2 ξ1 = – i
–
1. Then ξ = ( u
–
i – 1 ) ξ1 maps a1
to 1 and a2 to 0, which does not depend on H, note. Also 1 + ξ = 1 + ( u
–
i –
– 1 ) ξ1 = [ 1 + ξ1 , u
i
] ∈ H. In the same way H contains [ 1 + Y, v
j
] = 1 + Y ( v
j – 1 ).
Now v
j – 1 is a non-zero element of Zp and hence has an inverse w in Qp . Thus
Y w = Y and so Y = Y ( v
j – 1 ). If c ∈ C \ {0} there is an element η of Y with 1 η =
= c. (Note the wide choice for c available here; we will make more specific choices
for c later.) Then the periodic residual N =
H
H∩ of G contains 1 + ξ and 1 + η.
A simple calculation shows that [ 1 + ξ, 1 + η ] = 1 + ξ η ∈ 1 + Z and a1 ξ η = 1 η =
= c ≠ 0. Therefore N is not abelian.
Finitary groups over finite images of Z are locally finite (e.g. [5], 11). Thus by
3.4c) we have [ N , N , N ] = 〈 1 〉. We now show that in general (that is, for some
subgroups G of F∞ Aut M with periodic residuals N ) neither [ N, N, G ] nor [ N, G,
N ] nor N G N G, , ,[ ] [ ][ ] need be 〈 1 〉; in particular the periodic residual of G need
not be central (in G ) by abelian nor abelian by central (in G ).
We may choose c in the above construction with c v ≠ c. Then ξ η v ≠ ξ η. Since
[ 1 + ξ, 1 + η ]
diag
(
1,1,v
)= 1 + ξ η v, the element [ 1 + ξ, 1 + η ] does not lie in the
centre of G. Then with this choice of c we have [ N, N, G ] ≠ 〈 1 〉.
Now choose any c with | c | > 4, and we can do this with c v ≠ c if we wish.
Now
[ 1 + ξ, diag ( – 1, 1, – 1 ) ] = 1 – 2 ξ,
[ 1 + η, diag ( – 1, 1, – 1 ) ] = 1 – 2 η,
[ 1 – 2 ξ, 1 – 2 η ] = 1 + 4 ξ η
and
a1 ( 4 ξ η ) = 4 c ≠ 0.
Set G0 = 〈 G, diag ( – 1, 1, – 1 ) 〉. Since [ M, diag ( – 1, 1, – 1 ) ] = 2 A ⊕ C, so G 0 ≤
≤ F∞ Aut M. The periodic residual N0 of G0 contains N and
[ N0, G0, N0 ] ≥ N G N G0 0 0 0, , ,[ ] [ ][ ] ≥ N G N G, , ,0 0[ ] [ ][ ] ≠ 〈 1 〉.
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FINITARY GROUPS AND KRULL DIMENSION OVER THE INTEGERS 1319
Thus our second and third claims are also substantiated.
3.6. Lemma. Let G ≤ F∞ Aut M and set A = CG ( M / T ) ∩ CG ( T ). Then A
is abelian and unipotent and G / A is residually ( finitary over Q and finite
images of Z ).
Proof. The group G / CG ( M / T ) embeds into F G L ( Q ⊗ M ) and so is actually
finitary over Q . Also G / CG ( T ) is residually (finitary over finite images of Z ) by
3.4a). The claim follows.
3.7. Examples. In 3.6 we cannot always find a central subgroup of G modulo
which G is residually finitary.
For let M and G ≤ Aut M be as in 6.3 of [6]. Then M is minimax, so G ≤
≤ F∞ Aut M. Also G is finitely generated and has a central Prüfer subgroup, Z say.
If ρ : G → F AutR N is a homomorphism for some module N over any commutative
ring R, then G ρ is residually finite ([6], 6.2b)), so Z ρ = { 0 }. Thus G is not
residually (finitary over commutative rings). The same argument (and [6], 6.2c))
shows that G is not residually (Artinian-finitary over commutative rings) either. The
offending subgroup in both cases, namely Z, is central in G . The wreath product W
of G by a cyclic group of order 2 embeds into F∞ Aut M
(
2
) and the intersection of all
normal subgroups N of W with W / N either finitary of Artinian-finitary over some
commutative ring, is abelian and normal but not central in W.
3.8. Proof of Theorem 1. Part a) of Theorem 1 follows from 2.1b) & 3.2, Part b)
from 3.4 and [5], 11 (if M has finite exponent, then F∞ Aut M is always locally finite)
and Part c) from 3.6. The examples mentioned in the Introduction, preventing various
generalizations or extensions of Theorem 1, are given in 3.3, 3.5 and 3.7.
3.9. Proof of Theorem 3. Set B = CG ( M / T ) ∩ CG ( T ). Clearly B is abelian
and normal in G. Let T = ⊕p pT , where Tp is the p-primary component of T.
Then G / B embeds into
G C M T C TG p G p/ /( )( ) × × ( )( )( ),
and I do mean direct product here and not just cartesian product. G C M TG/ /( )
embeds naturally into F G L ( Q ⊗ M ) and G / CG ( Tp ) embeds naturally into
F1 Aut Tp . If g ∈ G, then g is unipotent if and only if the images of g in F G L ( Q ⊗
⊗ M ) and the F1 Aut Tp are unipotent. Thus it suffices to prove the result just in the
two cases where either M is torsion-free or M is a p-group for some prime p.
Suppose M is torsion-free, so G ≤ F G L ( Q ⊗ M ). Set K = 〈 1 〉 = B. Here u ( G )
is torsion-free. Thus the first part of c) is trivial in this case; the second part of c) is
given by 3.4b). Part d) here is trivial with Q the only field arising, as is Part e). Part
f), of course, is totally trivial.
Now assume that M is a p-group. Then M is a module in the obvious way over
the local ring of Z at p. Hence Section 7 of [9] is applicable. Let
V = M [ p ] = { x ∈ M : p x = 0 }
and set H = CG ( V ) and K = H ∩ u ( G ). By [9], 7.1, the group H is residually (a
nilpotent p-group of finite exponent) and G / H embeds naturally into the finitary
linear group F G L ( V ) over G L ( p ). Also u ( G ) is residually a p-group and H
stabilizes an ascending series in M, cf. the proof of [9], 7.3. We now have a), b), c) d)
and f). By b) of Theorem 2 (which we have already proved, see 2.2) the group
G / u ( G ) is a subdirect product of finitary linear groups over Q, G F ( p ) and Qp
(only for the one prime p ). Since G / K embeds into G / H × G / u ( G ), so Part e)
follows.
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1320 B. A. F. WEHRFRITZ
A similar argument yields the following.
3.10. Theorem. Let G be a subgroup of F∞ Aut M. Then G has a normal
series
〈 1 〉 ≤ B ≤ K ≤ s ( G ) ≤ G,
where
a) B is abelian,
b) K / B is residually (nilpotent of finite exponent),
c) s ( G ) is the s-radical of G and is locally residually nilpotent,
d) s ( G ) / B is a subdirect product of a torsion-free Fitting group and a
residually (periodic Fitting group); it is also abelian by residually (a periodic
Fitting group),
e) s ( G ) / K is a subdirect product of unipotent finitary linear groups over Q
and G F ( p ) for various primes p,
f) G / K is a subdirect product of finitary linear groups over Q and G F ( p )
for various primes p,
g) G / s ( G ) is a subdirect product of irreducible finitary linear groups over Q
and G F ( p ) for various primes p,
h) K stabilizes an ascending series in M.
Proof. Again set B = CG ( M / T ) ∩ CG ( T ). A torsion-free Z-module is tight, so
any s-subgroup of G acts unipotently on M / T by 4.3 of [2]. Thus again we may
assume that M is either torsion-free or a p-group for some prime p. If M is torsion-
free set K = 〈 1 〉. Here s ( G ) = u ( G ) and the claims follow easily in this case.
Now assume that M is a p-group. Set B = 〈 1 〉, V = M [ p ] and K = CG ( V ). Then
K satisfies h), cf. the proof of [9], 7.3, and b) by [9], 7.1. In particular K ≤ s ( G ).
The inverse image in G of the unipotent radical of G on V acts unipotently on Vi =
= { x ∈ M : p
i
x = 0 } for each i and clearly s ( G ) acts unipotently on V. Thus this
inverse image is s ( G ). Also G / K embeds into F G L ( V ) and the above shows that
s ( G ) / K maps onto the unipotent radical of this image of G / K. In particular this
yields e). Part c) is part definition and Part a) consequence of [2], 4.6. Also s ( G ) is
residually a Fitting p-group; this can be seen directly, alternatively cf. [9], 7.3. The
second part of d) is then trivial. G / s ( G ) is a subdirect product of irreducible finitary
linear groups by 4.1 of [6]. A check of proof of that shows that the only ground field
that arises in this case is G F ( p ). This proves g). Finally f) follows from the action of
G on V.
4. Unipotent subgroups.
4.1. Lemma. Let E be the intersection of all subgroups X of M with M / X
divisible and let G be a unipotent subgroup of Aut M.
a) E has no non-zero divisible image.
b) D ∩ E = { 0 } for D the divisible component of T.
c) G / CG ( M / E ) and G / CG ( D ) are torsion-free.
d) F∞ Aut E = F Aut E.
e) If G ≤ F∞ Aut M, then G / CG ( T / D ) and G / CG ( E ∩ T ) are periodic.
Proof. a) If E is divisible, then M = E ⊕ X for some X and then M / X is
divisible. Hence E ≤ X and E = { 0 }. Part a) follows from this.
b) Also M = D ⊕ X for some X, so M / X is divisible, E ≤ X and D ∩ E = { 0 }.
c) Let g ∈ G have finite order on M / E. Then M ( g – 1 ) modulo E has finite
exponent by [9], 2.1. Hence if M / X is divisible, then E ≤ X, so ( M / X ) ( g – 1 ) is
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FINITARY GROUPS AND KRULL DIMENSION OVER THE INTEGERS 1321
divisible of finite exponent and hence is { 0 }. Thus M ( g – 1 ) ≤ X
X∩ = E and g ≡
≡ 1 modulo E.
Now assume that g ∈ G has finite order on D. Then D ( g – 1 ) is divisible of
finite exponent and hence D ( g – 1 ) = { 0 }. We have now shown that G / CG ( M / E )
and G / CG ( D ) are torsion-free.
d) Let g ∈ F∞ Aut E. Then E / CE ( g ) � E ( g – 1 ) is minimax with (by a) no non-
zero divisible image. Hence E ( g – 1 ) is finitely generated and so g ∈ F Aut E . It
follows that F∞ Aut E = F Aut E.
e) Here G ≤ F∞ Aut M. Let g ∈ G. Then ( T / D ) ( g – 1 ) is a periodic minimax
group with zero divisible component. Thus ( T / D ) ( g – 1 ) is finite and consequently
g has finite order on T / D, e.g. by [9], 2.2c). Therefore G / CG ( T / D ) is periodic.
Finally G / CG ( E ∩ T ) is periodic by d) and [9], 2.2c) again.
4.2. Proof of Theorem 4a). Let P be the maximal periodic subgroup of G.
Using the notation of 4.1 we have from 4.1c) and [9], 2.2a) that
P ≤ CG ( M / E ) ∩ CG ( M / T ) = CG ( M / ( E ∩ T ) ).
Set Q = CP ( T / D ). Now [ E ∩ T, Q ] ≤ D ∩ E = { 0 } by 4.1b). Thus Q stabilizes
the series M ≥ E ∩ T ≥ { 0 } and consequently Q is periodic and abelian. Also G / Q
embeds into
( G / P ) × G C T DG/ /( )( ) ,
so G / Q is torsion-free by periodic, see 4.1e).
4.3. Example. Let M = Z ⊕ P, where P is the direct sum of two Prüfer p-
groups, for some prime p. Then M is minimax and it is easy to see that F∞ Aut M =
= Aut M contains a unipotent subgroup G isomorphic to the split extension of P by
the subgroup of Aut P generated by
1 0
1 1
. Further it is easy to see that G is not
torsion-free by periodic, so the abelian normal subgroup of Theorem 4a) cannot be
eliminated.
4.4. Proof of Theorem 4b). By Theorem 2c) we have U ∈ N and by 2.5a) (and
2.1a)) we have [ M, U ] ≤ T with U / CU ( T ) embeddable into F1 Aut T. We apply the
proof of [9], 4.4, to T. In doing so we construct a series
M ≥ T = N0 ≥ N1 ≥ … ≥ Nr ≥ N ≥ { 0 }
of finite length stabilized by U with T / Nr of finite exponent and N = n
nT∩ ; note
that [ N, U ] = { 0 } by 2.5a) again.
Set V = CU ( T / N ). Then V stabilizes the series M ≥ T ≥ N ≥ { 0 }, so V ∈ N2
.
Also U / V stabilizes the series N0 ≥ N1 ≥ … ≥ Nr ≥ N with N0 / Nr of finite
exponent. Thus U / V ∈ E and hence U ∈ N2
E.
Now set W = CU ( Nr ) ∩ CU ( M / Nr
). Here W stabilizes M ≥ Nr ≥ { 0 }, so W is
abelian. Similarly U / CU ( Nr ) is abelian since it stabilizes Nr ≥ N ≥ { 0 }. Finally
U / CU ( M / Nr
) stabilizes M ≥ N0 ≥ … ≥ Nr with N 0 / Nr
of finite exponent, so
U / CU ( M / Nr
) ∈ E . Consequently U / W ∈ S ( A × E ) and U ∈ A.S ( A × E ) .
Trivially then, U ∈ A E A.
4.4. Proof of Theorem 4c). This is actually a simple corollary of Theorem 4b).
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1322 B. A. F. WEHRFRITZ
If X is a finite subset of G, then 〈 X
G
〉 ∈ N ∩ N2
E ∩ A E A by Theorem 4b). If Y
is a finite subset of G ′, then Y lies in 〈 X 〉 ′ for some finite subset X of G. Hence
〈 Y
G
〉 ≤ 〈 X
G
〉 ′ and so 〈 Y
G
〉 ∈ N ∩ A E.
5. Maximin groups; a variant. Instead of minimax groups, suppose we consider
abelian groups that are extensions of Artinian groups by Noetherian groups. Such a
group is just
(*) a direct sum of a finite number of cyclic and Prüfer groups.
This class does not fit into the general framework of [2], but does still define a finitary
group (cf. [3], Section 2); namely
FA + N Aut M = { g ∈ Aut M : M ( g – 1 ) satisfies (*) above}.
The suffix A + N here is suggested by the fact that an abelian group satisfies (*) if and
only if it is a sum of an Artinian group and a Noetherian group. We coin the term
maximin group for such a group, a name that seems not to have been used before. (In
contrast the term minimax group has been in used for some forty years at least.)
Clearly 〈 F Aut M, F1 Aut M 〉 ≤ FA + N Aut M ≤ F∞ Aut M for any abelian group M.
Thus Theorems 1 to 4 trivially apply to F MA N+ Aut and its subgroups. Further the
counter examples constructed in 2.3, 3.5 and 4.3 all lie in FA + N Aut M. This in not the
case for the examples constructed in 3.3 and 3.7, which are based on the construction
of [6], 6.3. This raises the possibility that Parts a) and c) of Theorem 1 can be
strengthened for FA + N Aut M. In both cases they can.
Theorem 5. Let M be an abelian group with torsion subgroup T. Set G =
= FA + N Aut M.
a) A = CG ( M / T ) ∩ CG ( T ) is an abelian normal subgroup of G and G / A
embeds into the direct product F Aut ( M / T ) × F1 Aut T. There is a natural exact
sequence
1 → F1 Aut M → FA + N Aut M → F Aut ( M / T ).
b) G is locally residually finite.
c) G is unipotent-abelian by residually (finitary over Z ).
5.1. Lemma. Let G be a finitely generated subgroup of AutR M, where M
is a module over the commutative ring R. Suppose [ M, G ] ≤ N + A , where N is
a Noetherian R-submodule of M and A is an Artinian R-submodule of M. I f
X is a finite subset of M, then R X G =
x X g G
Rxg∈ ∈∑ ,
is finitely generated as
R-module.
Proof. If A+ denotes the sum of all the Artinian R-submodules of N + A, then
A ≤ A+ and A+ / A is R-Noetherian. It follows that A+ / A and hence A+ are R-
Artinian. Hence we may assume that A = A+ . Clearly then A G ≤ A . Define Ai
inductively by letting A0 = { 0 } and letting Ai + 1 / Ai denote the socle of the R-
module A / Ai . Then
i iA≥0∪ = A, each Ai is Noetherian (as well as Artinian) and
Ai G ≤ Ai for each i.
Suppose X = { x1 , x2 , … , xl }, N = R y1 + R y2 + … + R ym and G = { g1 , g2 , …
… , gn }. There exists an integer t such that for all i, j and k we have xi ( gk – 1 ) ∈
∈ N + At and yj ( gk – 1 ) ∈ N + At . In particular N gk ≤ N + At for each k and so ( N +
+ At ) G ≤ N + At . But then the g k and hence each element of G centralize R X
modulo N + At . Consequently R X G ≤ R X + N + At and so R X G = R X + ( ( N + At ) ∩
∩ R X G ). But N + At is Noetherian, so ( N + At ) ∩ R X G is too, and X is finite by
hypothesis. Therefore R X G is finitely generated.
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FINITARY GROUPS AND KRULL DIMENSION OVER THE INTEGERS 1323
5.2. Proof of Theorem 5.
a) As in the proof of 2.1 we have F1 Aut M = CG ( M / T ). Clearly A is abelian,
FA + N Aut ( M / T ) = F Aut ( M / T ) and F TA N+ Aut = F1 Aut T. Both claims in a)
follow from this.
b) Let H be a finitely generated subgroup of G and apply 5.1 to H with R = Z.
If X is any finite subset of M, then Z X H is finitely Z-generated. Hence
H / CH ( Z X H ) is residually finite. (In fact H / CH ( Z X H ) for every prime p is
(residually a finite p-group) by finite.) Part b) follows.
c) A is unipotent and abelian, trivially F Aut ( M / T ) is finitary over Z and
F1 Aut T is residually (finitary over finite images of Z ), e.g. by [9], 8.3. Part c)
follows.
If M itself is maximin, we can squeeze a little more out of this.
5.3. Remark. Let H be a finitely generated subgroup of G = Aut M, where M
is an abelian maximin group.
a) G need not be residually finite and need not be quasilinear.
b) H is isomorphic to a linear group of finite degree over the complex numbers C.
c) For all but a finite set of primes p the group H is (residually a finite p-group)
by finite.
Proof. a) If K is the split extension of a Prüfer p-group P by the group U of
units of the ring of p-adic integers, then K embeds into Aut ( Z ⊕ P ) and is not
residually finite and is not quasilinear, see [6], 5.2.
b) M = F ⊕ D, where F is finitely generated and D is divisible and Artinian.
Thus we may take Aut M in the form
Aut Hom
Aut
F F D
D
( , )
0
.
Hence H ≤ 〈 X, Y, Z0 〉, where X is a finitely generated subgroup of Aut F ⊕ 1D , Y is
a finitely generated subgroup of 1F ⊕ Aut D and Z 0 – 1 is a finite subset of
Hom( , )F D . There exists an integer n such that if Dn = { x ∈ D : n x = 0 }, then Z0 –
– 1 lies in Hom( , )F Dn . Set Z = 1 + Hom( , )F Dn . Then Z is a finite normal
subgroup of H 0 = X Y Z and H ≤ H0 . Clearly ( H0 : X Y ) is finite and X Y is
isomorphic to a linear group of finite degree over C, consequently so too are H0 and
H.
c) This follows from Part b) and 4.7 of [11].
Suppose M is torsion-free. Then F∞ Aut M is finitary over Q (see 2.1) and
abelian by (residually finitary over finite images of Z ) by 3.4b). Further F MA N+ Aut
is actually finitary over Z. However F∞ Aut M need not be finitary over Z or even
residually (finitary over Z ) as we show in 5.6 below. In general, that is with M not
necessarily torsion-free, while F∞ Aut M is (nilpotent of class ≤ 2) by residually
(finitary over finite images of Z ) by 3.4c) and while F MA N+ Aut is abelian by
residually (finitary over Z ) by Theorem 5, F∞ Aut M need not be abelian by residually
(finitary over Z ). This too we show below, see 5.7. We need a little preparation.
5.4. Example. Let G = a b a ab q, = , where q is some prime. Then G is
isomorphic to the subgroup
1 0
1 1
0
0 1
,
q
of GL q2 1, Z /[ ]( ). Thus if M1 is the direct product of two Prüfer p-groups for any
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1324 B. A. F. WEHRFRITZ
prime p ≠ q, then G embeds into FA + N Aut M1 = F1 Aut M1 = Aut M1
. If M2 is a
free Z 1/[ ]q -module of rank 2, then M2 is Z-torsion-free and G embeds into
F∞ Aut M2 = Aut M2
.
However G does not embed into F Aut M for any abelian group M, for if it did
we would have G acting faithfully on some finitely generated subgroup of M by [4],
2.3c), so G would embed into G L ( n, Z ) for some integer n . But G L ( n , Z ) is
(residually a finite q-group) by finite, while G is not; indeed its subgroup 〈 a
G
〉 is
isomorphic to the additive group of Z 1/[ ]q . However G is residually finite and
hence is trivially residually (finitary over finite images of Z ).
5.5. Example. Let p ≠ q be primes and let G = 〈 g 〉 P be the extension of the
Prüfer p-group P by its automorphism g : h � h
q
. Then G does not embed into
F Aut M for any abelian group M.
Proof. Suppose G ≤ F Aut M. Then [ M, P ] ≤ D ≤ T and [ T, P ] = { 0 } by
2.5b). Hence N = T ∩ M ( g – 1 ) is finite and centralized by P. Thus N is a G -
submodule of M and C M NP /( ) is finite. Therefore P C M NP/ /( ) is isomorphic to
P and so G C M NG/ /( ) is isomorphic to G. Consequently we may pass to M N/
and assume that N = { 0 }. In particular now T ( g – 1 ) = { 0 } and so [ T, G ] = { 0 }.
Let x ∈ M and h ∈ P. Then h
g
= h
q
and so
x g
–
1
( h – 1 ) = x ( h
g
– 1 ) = x ( h
q
– 1 ) = q x ( h – 1 ),
where we have used [ M, P ] ≤ T and [ T, G ] = { 0 }. Thus x g
–
1 is congruent to q x
modulo CM ( h ) for any h in P, so in fact x g
–
1 is congruent to q x modulo CM ( P ).
Set M
* = M / CM ( P ). Then ( q – 1 ) M
* = M
*
( g
–
1 – 1 ), which is finitely
generated. Also M
* and hence ( q – 1 ) M
* is q-divisible. Thus ( q – 1 ) M
* is finite
(even a finite q ′-group). Hence M
* has finite exponent (prime to q ). But P
stabilizes the series M ≥ CM ( P ) ≥ { 0 }, so P embeds into the group Hom ( M
*,
CM ( P ) ), which also has finite exponent. This is manifestly false and 5.5 follows.
5.6. Example. Let G = A b a a a Ab q, = for all in , where A is a copy of
the additive group of Q and q is a prime.
a) G embeds into F q∞ /[ ] ⊕( )Aut Z Q1 .
b) G is not residually periodic; indeed A ≠ 〈 1 〉 is the periodic residual of G.
c) G is not residually (finitary over Z ) and A is the (finitary over Z ) residual
of G.
Note that in a) the abelian group Z Q1/[ ] ⊕q is torsion-free. Trivially both b)
and c) imply that G is not residually (finitary over finite images of Z ).
Proof. a) Clearly G is isomorphic to the subgroup
q
r
r0
0 1
1
0 1
∈, :
φ
Q
of F q∞ /[ ] ⊕( )Aut Z Q1 , where φr denotes the homomorphism of Z 1/[ ]q into Q
given by 1 � r.
b) Suppose N is a normal subgroup of G with G / N periodic and A not
contained in N. Then A N / N is nontrivial, periodic and divisible. Also b acts on
A N / N by raising elements to their q-th powers. Hence G / N contains a copy of the
group G of 5.5 for some prime p ≠ q. This group is not periodic. It follows easily
that A lies in, and hence is, the periodic residue of G.
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FINITARY GROUPS AND KRULL DIMENSION OVER THE INTEGERS 1325
c) Now suppose that N is the kernel of a homomorphism of G into some
F Aut M. Assume 〈 1 〉 ≠ A ∩ N ≠ A . Then again A N / N is periodic and G / N
contains a copy of the group G of 5.5 for some prime p ≠ q. This contradicts 5.5 and
therefore either A ∩ N = 〈 1 〉 or A ≤ N.
Suppose A ∩ N = 〈 1 〉. Since CG ( A ) = A, this implies that N = 〈 1 〉. But G
contains a copy of the group G of 5.4. This contradicts the conclusion of 5.4 that its
group embeds into no finitary group over Z. Therefore A ≤ N. It follows easily that A
is the (finitary over Z ) residual of G.
5.7. Example. Let M = Z 1/[ ]q ⊕ B ⊕ C, where B is the additive group of Q ,
where C is a Prüfer p-group and where p and q are distinct primes. Set
G =
q X Z
Y
q
0 1
0 0
≤ F∞ Aut M,
where X = Hom Z 1/[ ]( )q B, � B, Y = Hom ( B, C ) � Qp and Z = Hom Z 1/[ ]( )q C,
� C. Note that q is a unit of Z p = End C . Let N denote the (finitary over Z )
residual of G. Then N contains 〈 1 + X, 1 + Y 〉. In particular G is not abelian by
residually (finitary over Z ).
Proof. Firstly it is easy to check that G lies in F∞ Aut M. Now apply 5.6c) to the
groups
q X
0 1
and
1
0
Y
q
.
We deduce that N contains 1 + X and 1 + Y. If ξ denotes the natural embedding of
Z 1/[ ]q into B = Q and if η is any homomorphism of B into C with 1 η ≠ 0, then
1 ξ η ≠ 0 and ξ η ≠ 0. But [ 1 + ξ, 1 + η ] = 1 + ξ η ≠ 1. Therefore N is not abelian.
(Necessarily by 3.4c) N is nilpotent of class 2.)
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Received 10.03.2005
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