On groups with the minimal condition for non-invariant decomposable abelian subgroups
The infinite groups, in which there is no any infinite descending chain of non-invariant decomposable abelian subgroups (md-groups) are studied. Infinite groups with the minimal condition for non-invariant abelian subgroups, infinite groups with the condition of normality for all decomposable abel...
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Lyman, F.N. Drushlyak, M.G. 2019-06-20T03:08:20Z 2019-06-20T03:08:20Z 2006 On groups with the minimal condition for non-invariant decomposable abelian subgroups / F.N. Lyman, M.G. Drushlyak // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 57–66. — Бібліогр.: 3 назв. — англ. 1726-3255 https://nasplib.isofts.kiev.ua/handle/123456789/157375 The infinite groups, in which there is no any infinite descending chain of non-invariant decomposable abelian subgroups (md-groups) are studied. Infinite groups with the minimal condition for non-invariant abelian subgroups, infinite groups with the condition of normality for all decomposable abelian subgroups and others belong to the class of md-groups. The complete description of infinite locally finite and locally soluble non-periodic md-groups is given, the connection of the class of md-groups with other classes of groups are investigated. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On groups with the minimal condition for non-invariant decomposable abelian subgroups Article published earlier |
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On groups with the minimal condition for non-invariant decomposable abelian subgroups |
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On groups with the minimal condition for non-invariant decomposable abelian subgroups Lyman, F.N. Drushlyak, M.G. |
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On groups with the minimal condition for non-invariant decomposable abelian subgroups |
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On groups with the minimal condition for non-invariant decomposable abelian subgroups |
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On groups with the minimal condition for non-invariant decomposable abelian subgroups |
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On groups with the minimal condition for non-invariant decomposable abelian subgroups |
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on groups with the minimal condition for non-invariant decomposable abelian subgroups |
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Lyman, F.N. Drushlyak, M.G. |
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Lyman, F.N. Drushlyak, M.G. |
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Algebra and Discrete Mathematics |
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The infinite groups, in which there is no any infinite descending chain of non-invariant decomposable abelian subgroups (md-groups) are studied. Infinite groups with the minimal
condition for non-invariant abelian subgroups, infinite groups with
the condition of normality for all decomposable abelian subgroups
and others belong to the class of md-groups. The complete description of infinite locally finite and locally soluble non-periodic
md-groups is given, the connection of the class of md-groups with
other classes of groups are investigated.
|
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1726-3255 |
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https://nasplib.isofts.kiev.ua/handle/123456789/157375 |
| citation_txt |
On groups with the minimal condition for non-invariant decomposable abelian subgroups / F.N. Lyman, M.G. Drushlyak // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 57–66. — Бібліогр.: 3 назв. — англ. |
| work_keys_str_mv |
AT lymanfn ongroupswiththeminimalconditionfornoninvariantdecomposableabeliansubgroups AT drushlyakmg ongroupswiththeminimalconditionfornoninvariantdecomposableabeliansubgroups |
| first_indexed |
2025-11-25T22:45:10Z |
| last_indexed |
2025-11-25T22:45:10Z |
| _version_ |
1850570707113607168 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2006). pp. 57 – 66
c© Journal “Algebra and Discrete Mathematics”
On groups with the minimal condition for
non-invariant decomposable abelian subgroups
F. N. Lyman and M. G. Drushlyak
Communicated by L. A. Shemetkov
Abstract. The infinite groups, in which there is no any infi-
nite descending chain of non-invariant decomposable abelian sub-
groups (md-groups) are studied. Infinite groups with the minimal
condition for non-invariant abelian subgroups, infinite groups with
the condition of normality for all decomposable abelian subgroups
and others belong to the class of md-groups. The complete de-
scription of infinite locally finite and locally soluble non-periodic
md-groups is given, the connection of the class of md-groups with
other classes of groups are investigated.
The subgroup H of the group G will be called decomposable if it
decomposes in the direct product of two non-trivial factors.
Infinite groups, in which there is no any infinite descending chain of
non-invariant decomposable abelian subgroups, are studied in this article.
Such groups will be called md-groups. Thus, if the md-group G contains
infinite descending chain of decomposable abelian subgroups A1 ⊃ A2 ⊃
... ⊃ An ⊃ ..., then the subgroup Ak is certainly invariant in the group
G for some natural number k.
The class of md-groups is wide enough. For example, infinite groups
with such restrictions as the minimal condition for non-invariant abelian
subgroups (Chernikov’s I-groups [1]); the condition of normality for all
infinite abelian subgroups (Chernikov’s IH-groups [1]); the condition of
normality for all decomposable abelian subgroups (in the case of non-
abelian group such groups are called di-groups).
Key words and phrases: group, subgroup, order of the group, involution, locally
finite group, non-periodic group, decomposable abelian subgroup, minimal condition,
condition of normality.
58 On groups with the minimal condition
All infinite groups, in which every subgroup does not decompose in
the direct product of two non-trivial factors, concern to the md-groups.
Such md-groups are described in [2] with condition of locally fineness in
the periodic case and with the condition of locally solubility in the non-
periodic case. From the theorems 1.1 and 2.1 of the article [2] will have
following propositions, which concern md-groups without decomposable
subgroups.
Theorem 1. In an infinite locally finite group G all abelian subgroups
are indecomposable if and only if it is the group of the one of the following
types:
1) G is quasicyclic p-group for some prime p;
2) G = A 〈b〉, where A is quasicyclic 2-group, |b| = 4, b2 ∈ A and
b−1ab = a−1 for every element a ∈ A;
3) G = Aλ 〈b〉 is the Frobenius’ group, where A is quasicyclic p-group,
B is cyclic q-group, p and q are prime, (p − 1, q) = q.
Theorem 2. In a locally soluble non-periodic group G all abelian sub-
groups are indecomposable if and only if it is the group of one of the
following types:
1) G is abelian torsion free group of rang 1;
2) G = AλB is the Frobenius’ group, where A is abelian torsion free
group rang 1, |b| = 2 or |b| = ∞.
Since the periodic indecomposable abelian group is cyclic or quasi-
cyclic p-groups for some prime p, then the minimal condition for non-
invariant decomposable abelian subgroups is equal to the minimal con-
dition for non-invariant abelian subgroups in the periodic case. Groups
with such restrictions were studied by Chernikov (see [1], theorem 4.11).
That is why only non-periodic md-groups are studied further.
Theorem 3. A non-periodic group G, which does not satisfy the min-
imal condition for decomposable abelian subgroups, satisfies the minimal
condition for non-invariant decomposable abelian subgroups if and only if
it is the group of one of the following types:
1) G is non-periodic abelian groups with decomposable subgroups;
2) G = Q×B, where Q is the quaternion group of order 8, B is abelian
torsion free group of rank 1;
F. N. Lyman, M. G. Drushlyak 59
3) G = 〈x〉λA, where |x| = pn, p is prime (p = 2, n > 1), A is non-
divisible abelian torsion free group of rank 1 and commutant G′ is
of prime order;
4) G =
(〈
x2
〉
× A
)
〈x〉, where |x| = 8, A is abelian torsion free group
of rank 1 and quotient group G/
〈
x4
〉
is IH-group;
5) G = (〈z〉 × A) λ 〈x〉, where |z| = 4, |x| = 2, A is abelian torsion
free group of rank 1 and quotient group G/
〈
z2
〉
is IH-group;
6) G = A 〈b〉, where A is non-periodic abelian group, b4 = 1 and
b−1ab = a−1 for an arbitrary element a ∈ A, the centre Z(G) is of
order 2n, n ≥ 0 and when |b| = 2 the group A contains decomposable
subgroups;
7) G = Aλ 〈b〉, where A is non-periodic abelian group, which does not
contain free abelian subgroups of rank 2, or non-abelian di-group
with infinite centre, |b| = 2, the torsion part T (A) is 2-group with
one involution 〈a〉 and G/ 〈a〉 is IH-group;
8) G = Cλ 〈x〉, where C = CG (a), |a| = ∞, 〈a〉 ⊳ G, |Z (G)| = p and
G/Z(G) - IH-group;
9) G = (〈b〉λA) λ 〈x〉, where |b| = p 6= 2, A = 〈a〉A1 is abelian torsion
free group of rank 1, a2 ∈ A1, [A1, b] = [x, b] = 1, a−1ba = b−1,
x−1yx = y−1 for an arbitrary element y ∈ A.
Proof. It is evident that the groups of every type which is enumerated
in the condition of the theorem satisfy the minimal condition for non-
invariant decomposable abelian subgroups. That is why we will prove
only the necessity of the conditions of the theorem. The proof of necessity
is completed by lemmas, which are proved further.
Lemma 1. If a non-periodical abelian subgroup A of the md-group G con-
tains free abelian subgroups of rank 2 or decomposable periodic subgroup,
then all subgroup of A are invariant in the group G. If A = 〈a〉 × 〈b〉,
where |a| = ∞, 1 < |b| < ∞, then 〈a, b〉 ⊳ G, all periodic subgroups of A
are invariant in the group G.
Proof. Suppose that A ⊃ 〈a〉 × 〈b〉, where |a| = ∞. Hence according
to the definition of the md-group the invariant subgroup
〈
apm
, b
〉
exists in
the group G among the subgroup of the chain 〈a, b〉 ⊃ 〈ap, b〉 ⊃
〈
ap2
, b
〉
⊃
... ⊃
〈
apn
, b
〉
⊃ ..., where p is prime. Also the invariant subgroup
〈
aqk
, b
〉
(q 6= p) exists in the group G. Since
(
pm, qk
)
= 1, then there are two
60 On groups with the minimal condition
integer numbers u and v, such as pmu+ qkv = 1. Then
〈
apm
, aqk
〉
= 〈a〉,
〈
apm
, b
〉
·
〈
aqk
, b
〉
= 〈a, b〉 ⊳ G.
If 1 < |b| < ∞, 〈b〉 ⊳ G comes from the condition 〈a, b〉 ⊳ G. Thus, all
periodic subgroups of the mixed abelian subgroup A of the md-group G
and it itself are invariant in G. Let A ⊃ 〈c〉 × 〈d〉, where 〈c〉 and 〈d〉 are
finite non-trivial subgroups. Then 〈y, c〉 ⊳ G, 〈y, d〉 ⊳ G for an arbitrary
element y ∈ A and that is why 〈y, c〉 ∩ 〈y, d〉 = 〈y〉 ⊳ G.
Let A ⊃ 〈a, b〉, where |a| = |b| = ∞. Then there is infinite cyclic
subgroup 〈a1〉 ⊂ 〈a, b〉 for an arbitrary element y ∈ A such as 〈a1〉∩〈y〉 =
1. Then according to proved earlier we have 〈an
1
, y〉 ⊳ G for every natural
number n. Hence ∩∞
n=1
〈an
1
, y〉 = 〈y〉 ⊳ G. Thus, all subgroups of A are
invariant in the group G, if the subgroup A contains the direct product
of two finite or two infinite cyclic subgroups. This completes the proof of
the lemma.
Lemma 2. All infinite cyclic subgroups are invariant in the non-periodic
non-abelian group G if and only if G = C 〈b〉, where C is non-periodic
abelian subgroup, b4 = 1 and b−1ab = a−1 for an arbitrary element a ∈ C.
Proof. This lemma was given without proof in the article [3]. Let us
pay attention at the necessary conditions of the lemma, because theirs
sufficiency is evident. Let G be non-periodic non-abelian group, in which
all infinite cyclic subgroups are invariant. Let us show, that the cen-
tralizer C of an arbitrary elemen x of infinite order of the group G is
abelian subgroup. If y ∈ C, z ∈ C, |y| = |z| = ∞, then 〈y〉 ⊳ G, 〈z〉 ⊳ G
and that is why [y, z] = 1. If c ∈ C and |c| < ∞, then |xc| = ∞
and [y, xc] = [y, c] = 1. If c1 ∈ C, c2 ∈ C and |c1| < ∞, |c2| < ∞,
then |c1x| = |c2x| = ∞ and [c1x, c2x] = [c1, c2] = 1. Consequently, the
subgroup C is abelian.
It is evident, that the subgroup C contains all elements of infinite
order and all elements of finite odd order of the group G. That is why
G = C 〈b〉, b2
n
= 1, n ≥ 1 and the element b is not commutative with
every element of infinite order of the group G. Thus, if y ∈ C, |y| =
∞, then b−1yb = y−1. If c ∈ C and |c| < ∞, then |cy| = ∞ and
b−1(cy)b = (cy)−1 = c−1y−1, b−1cb = c−1. At last, from the equalities
b−1(yb2)b = (yb2)−1 = y−1b2 we get b4 = 1. This completes the proof of
the lemma.
The corollary of this lemma is the Chernikov’s theorem (see [1], the-
orem 4.6) on the structure of non-periodic IH-groups - non-abelian non-
periodic groups, in which all infinite abelian subgroups are invariant.
Corollary. Non-periodic non-abelian group G is IH-group if and only if
it has the finite centre of the expornent, which is not more than 2, and
all its infinite cyclic subgroups are invariant.
F. N. Lyman, M. G. Drushlyak 61
Lemma 3. A non-periodic md-group G is abelian if its centre contains
the free abelian subgroup of rank 2 or the decomposable abelian subgroup
of order pq 6= 4,where p and q are prime.
Proof. Let the centre Z(G) of md-group G contains the free abelian
subgroup 〈a〉 × 〈b〉. Then all subgroups from 〈a, b, g〉 are invariant in G
for an arbitrary element g ∈ G by lemma 1. That is why 〈g〉 ⊳ G and the
group G is abelian.
Let Z(G) ⊃ 〈c〉 × 〈d〉, where the elements c and d are of finite order
and if only one of them is of odd order. Then by lemma 1 we have
〈y, c〉 ⊳ G, 〈y, d〉 ⊳ G for an arbitrary element y ∈ G of infinite order and
hence 〈y〉 ⊳ G. Thus, all infinite cyclic subgroups are invariant in the
group G. Since expZ(G) > 2, then the group G is abelian by the lemma
2. This completes the proof of the lemma.
Lemma 4. If a md-group G contains the periodic abelian subgroup A,
which does not satisfy the minimal condition, then all subgroups of A are
invariant in the group G.
Proof. According to the condition of the lemma, the subgroup A
contains the direct product A1 = 〈a1〉 × 〈a2〉 × ... × 〈an〉 × ... of the in-
finite quantity of subgroups 〈ai〉, i = 1, 2, ..., n, ... of finite order. Let
〈x〉 be an arbitrary cyclic subgroup of A. Then there are such infinite
subgroups B ⊂ A1, C ⊂ A1, that 〈x〉 ∩ B = 〈x〉 ∩ C = B ∩ C = 1. Let
B = 〈b1〉×〈b2〉×...×〈bn〉×..., C = 〈c1〉×〈c2〉×...×〈cn〉×.... Then accord-
ing to the definition of the md-groups every chain 〈x, b1, b2, ..., bn, ...〉 ⊃
〈x, b2, ..., bn, ...〉 ⊃ ..., 〈x, c1, c2, ..., cn, ...〉 ⊃ 〈x, c2, ..., cn, ...〉 ⊃ ... contains
the invariant subgroup. Let it be respectively subgroups 〈x, bk, ..., bn, ...〉
and 〈x, cm, ..., cn, ...〉. Their intersection is equal to 〈x〉 and that is why
〈x〉 ⊳ G. Hence all subgroups of A are invariant in G. This completes the
proof of the lemma.
Lemma 5. A non-abelian md-group G is the group of one of the types
2)-3) of the theorem 3, if it contains decomposable subgroups and has
non-periodic centre.
Proof. Let z ∈ Z(G), |z| = ∞ and B be an arbitrary decomposable
abelian subgroup of the group G. If the subgroup B is non-periodic, then
B⊳G by the lemma 1. If the subgroup B is periodic, then (〈z〉×B)⊳G by
the lemma 1 and hence B ⊳G. Thus, all decomposable abelian subgroups
are invariant in the group G. From the description of such groups in [2]
we have that the group G is the group of the one of the following types
2)-3) of the theorem 3. This completes the proof of the lemma.
Lemma 6. A non-periodic md-group G, which does not satisfy the mini-
mal condition for decomposable abelian subgroups, contains the invariant
infinite cyclic subgroup. The centralizer of every infinite cyclic invariant
subgroup contains all elements of infinite order.
62 On groups with the minimal condition
Proof. Let a non-periodic md-group G does not satisfy the minimal
condition for decomposable abelian subgroups. Then G has finite chain
A1 ⊃ A2 ⊃ ... ⊃ An ⊃ ... of abelian decomposable subgroups. There is
the subgroup Ai ⊳ G according to the definition of the md-group. Since
Ai does not satisfy the minimal condition, then the group G contains the
invariant infinite cyclic subgroup by lemmas 1 and 4.
Let 〈a〉 ⊳ G, |a| = ∞, x ∈ G, |x| = ∞. Let us show that x ∈ CG(a).
Suppose, that [a, x] 6= 1. Then x−1ax = a−1, 〈x〉 ∩ 〈a〉 = 1, x2 ∈ CG(a).
By the lemma 1 all subgroups from
〈
a, x2
〉
are invariant in the group
G. Then
[
x, ax2
]
= [x, a] 6= 1 and that is why x−1(ax2)x = (ax2)−1 =
a−1x2. Hence a−1x−2 = a−1x2, x4 = 1. We obtain a contradiction. Thus
[a, x] = 1. This completes the proof of the lemma.
Further let us investigate a non-periodic md-groups with the periodic
centre.
Lemma 7. If a non-periodic md-group G has the periodic centre Z(G) of
even order, then G = C 〈x〉, where C is the centralizer of every invariant
infinite cyclic subgroup of G, x8 = 1. expZ(G) = 2m ≤ 4 and |Z(G)| <
∞. If expZ(G) = 4, then Z(G) is the cyclic group.
Proof. By the lemma 6 the group G has invariant infinite cyclic sub-
group 〈a〉. Then G = C 〈x〉, where C = CG(a), x /∈ C, x2 ∈ C, |x| < ∞.
|x| = 2n, n ≥ 1 and x−1ax = a−1. Let |x| > 2. By the lemma 1
the subgroup
〈
ax2
〉
×
〈
x2
n−1
〉
is invariant in G. Then
〈
ax2, x2
n−1
〉2
=
〈
a2x4
〉
⊳ G. Hence x−1(a2x4)x = (a2x4)−1 = a−2x−4 = a−2x4, x−4 = x4,
x8 = 1.
Let us find out, elements of what order can be in the centre Z(G).
Let z ∈ Z(G), |z| < ∞. According to the condition of the lemma Z(G)
contains the involution i. Then (〈az〉 × 〈i〉) ⊳ G,
〈
a2z2
〉
⊳ G, x−1a2z2x =
(a2z2)−1 = a−2z2, z4 = 1.
We have, that |Z(G)| < ∞, because else Z(G) 〈x〉 is infinite abelian
2-group, which does not satisfy the minimal condition. Then we have
〈x〉 ⊳ G by the lemma 4. Hence we obtain the contradiction according to
the condition of the lemma.
Let expZ(G) = 4. Let us show, that in this case Z(G) = 〈z〉. Sup-
pose, that Z(G) ⊃ 〈z〉×〈c〉, where |z| = 4, |c| = 2. Then (〈az〉×
〈
z2
〉
)⊳G,
(〈az〉 × 〈c〉) ⊳ G. Hence
〈
az, z2
〉
∩ 〈az, c〉 = 〈az〉 ⊳ G. Then x−1(az)x =
(az)−1 = a−1z−1 = a−1z. And z2 = 1, we obtain the contradiction ac-
cording to the condition of the lemma. This completes the proof of the
lemma.
Lemma 8. If a non-periodic md-group G has the cyclic centre of order
4, then G is the group of the one of the types 4)-5)of the theorem 3.
Proof. By the lemma 7 we have G = C 〈x〉, where C = CG(a),
F. N. Lyman, M. G. Drushlyak 63
|a| = ∞, 〈a〉 ⊳ G, x8 = 1. Let |x| = 8. Then x2 ∈ C. Let us show,
that Z(G) =
〈
x2
〉
. Since expZ(G) = 4, then the group G contains the
non-trivial infinite cyclic subgroup by the lemma 2. The fact, that the
subgroup C does not contains the free abelian subgroup of rank 2 and
the decomposable periodic subgroup, follows from the lemma1. That is
why the torsion part T (C) is locally cyclic 2-group or the quaternion
group and all subgroups of T (C) are invariant in G. Since
〈
y, x4
〉
⊳ G
for an arbitrary element y ∈ G of infinite order, then all infinite cyclic
subgroups are invariant in quotient group G/
〈
x4
〉
and its centre is finite.
Hence G/
〈
x4
〉
is IH-group.
Let us specify the structure of the subgroup C. Let y1, y2 are arbitrary
elements of infinite order of the group G. Then
〈
y1, y2, x
4
〉
is abelian
group, because
〈
y1, y2, x
4
〉
/
〈
x4
〉
is cyclic group. Hence C is abelian
group and C =
〈
x2
〉
× A, where A is abelian torsion free group of rank
1. The group G is the group of the type 4) of the theorem 3.
Let Z(G) = 〈z〉, |z| = 4 and |x| ≤ 4. If |x| = 4, then 〈x〉 ∩ 〈z〉 6=
1, else all infinite cyclic subgroups are invariant in G. We obtain the
contradiction according to the condition of the lemma. Then (xz)2 = 1
and let us take the element xz instead the element x. Hence, |x| = 2.
Also C = 〈z〉 × A, where A is abelian torsion free group of rank 1. The
group G is the group of the type 5) of the theorem 3. This completes the
proof of the lemma.
Further let us study non-periodic md-groups, which have expZ(G) =
2 and 4 ≤ |Z(G)| < ∞.
Lemma 9. If the centre of non-periodic md-group G is finite elementary
2-subgroup of the order, which is not less then 4, then G is IH-group and
belongs to the groups of the type 6) of the theorem 3.
Proof. Let Z(G) ⊃ 〈z1〉 × 〈z2〉, |z1| = |z2| = 2, y ∈ G and |y| = ∞.
Then 〈y, z1〉⊳G, 〈y, z2〉⊳G by lemma 1. Hence 〈y〉⊳G. Since |Z(G)| < ∞,
then G is IH-group and according to the corollary of the lemma 2 G is
the group of the type 6) of the theorem 3. This completes the proof of
the lemma.
Lemma 10. If a non-periodic md-group G = C 〈x〉, where C = CG(a),
|a| = ∞ and 〈a〉 ⊳ G, |x| ≤ 4, |Z(G)| = 2, then either G is IH-group or
G/Z(G) is IH-group. G is the group of either the type 6) or the type 7)
of the theorem 3.
Proof. If the subgroup C contains the free abelian subgroup of rank 2
or the decomposable periodic subgroup, then all infinite cyclic subgroups
of the group G are invariant. Since |Z(G)| = 2, then G is IH-group
according to the corollary of the lemma 2 and G belongs to the groups
of the type 6) of the theorem 3.
Let the torsion part T (C) does not contain any decomposable sub-
64 On groups with the minimal condition
groups and C does not contains free abelian subgroups of rank 2. Then
T (C) is 2-group with one involution and that is why it is either the lo-
cally cyclic 2-group or the quaternion group. It is either 〈y〉 ⊳ G for an
arbitrary element y ∈ G of infinite order or there is such element y ∈ G,
that |y| = ∞, 〈y〉 is not invariant in G, but 〈y, i〉 ⊳ G, where 〈i〉 = Z(G).
In the first case we also have, that G is the group of the type 6) of
the theorem 3.
In the other case the quotient group Ḡ = G/ 〈i〉 is IH-group. All
decomposable subgroups of the group G are non-periodic, are contained
in C, are invariant in G, if T (C) 〈x〉 has one involution.
Suppose, that T (C) 〈x〉 contains the elementary abelian subgroup A
of order 4. Then, when |x| = 4 G ⊃ 〈x〉 × 〈i〉, where i2 = 1, we have
〈i〉 = Z(G). Since
[
x2, a
]
= 1, then
〈
x2
〉
⊳ G, x2 ∈ Z(G), and we obtain
a contradiction. Hence |x| = 2 and G is the group of the type 7) of the
theorem 3. This completes the proof of the lemma.
Lemma 11. If a non-trivial centre Z(G) of the non-periodic md-group
G is periodic subgroup with the elements of odd order, then |Z(G)| = p,
p is prime odd number.
Proof. The centre Z(G) is the locally cyclic p-group for some odd
prime p according to the lemma 3. Suppose, that Z(G) ⊃ 〈b〉, |b| = p2.
Then we have 〈ab, bp〉 ⊳ G, 〈ap, bp〉 ⊳ G, x−1apbpx = (apbp)−1 = a−pbp,
b−p = bp, b2p = 1 for subgroup 〈a〉, where |a| = ∞ and 〈a〉⊳G, x /∈ CG(a).
We obtain a contradiction. Thus, |Z(G)| = p. This completes the proof
of the lemma.
Lemma 12. If the centre Z(G) of the non-periodic md-group G is of the
order p 6= 2, then G = Cλ 〈x〉, |x| = 2, C = CG(a), |a| = ∞, 〈a〉 ⊳ G,
G/Z(G) is IH-group and the group G belongs to the groups of the type
8) of the theorem 3.
Proof. 〈y, Z(G)〉 ⊳ G for an arbitrary element y ∈ G of infinite order.
Then all infinite cyclic subgroups are invariant in Ḡ = G/Z(G). Since
Ḡ = C̄λ 〈x̄〉 and C̄ has not involution, then Z(Ḡ) = 1, Ḡ is IH-group.
The group G is the group of the type 8) of the theorem 3. This completes
the proof of the lemma.
Let us investigate the structure of the non-periodic md-groups, which
does not satisfy the minimal condition for decomposable abelian sub-
groups and has trivial centre. In this case we have G = Cλ 〈x〉, where
C = CG(a), |a| = ∞, 〈a〉⊳G, |x| = 2 and the subgroup C is either abelian
or di-group.
Lemma 13. If a non-periodic md-group G with the trivial centre does
not satisfy the minimal condition for the decomposable abelian subgroups
and G = Cλ 〈x〉, where C = CG(a) is abelian group, |a| = ∞, 〈a〉 ⊳ G,
|x| = 2, then G is IH-group and belongs to the groups of the type 6) of
F. N. Lyman, M. G. Drushlyak 65
the theorem 3.
Proof. If the subgroup C contains the decomposable abelian either
torsion free subgroup or periodic subgroup, then all subgroups of C are
invariant in G by the lemma 1. Since Z(G) = 1, then G is IH-group and
belongs to the groups of the type 6) of the theorem 3.
Suppose, that C does not contain decomposable torsion free sub-
groups and periodic decomposable subgroups. Then C = A×B, where A
is the abelian torsion free group of rank 1, B is the locally cyclic p-group,
p 6= 2. Let b ∈ B, |b| = p. According to the proved in the lemma 1 we
have B ⊳G and 〈b, y〉 ⊳G for an arbitrary element y ∈ G of infinite order.
Let 〈y〉 is not invariant in G. Then x−1yx = ykbm, x−1ypx = ypk = y−p.
Hence k = −1. Since [x, b] 6= 1, then x−1bx = bn, n 6= 1. b = x−2bx2 =
(x−1bx)n = bn2
. That is why n2 ≡ 1(modp) and n = −1, x−1bx = b−1.
Then y = x−2yx2 = x−1(y−1bm)x = (x−1yx)−1(x−1bx)m = y−kb−m,
b−m = yb−2m.
Hence b−2m = 1, bm = 1 and x−1yx = y−1. Thus, 〈y〉 ⊳ G. Hence
all infinite cyclic subgroups are invariant in G and Z(G) = 1. Then G
is IH-group and belongs to the groups of the type 6) of the theorem 3
according to the corollary of the lemma 2. This completes the proof of
the lemma.
Lemma 14. If a non-periodic md-group G with the trivial centre does
not satisfy the minimal condition for decomposable abelian subgroups and
G = Cλ 〈x〉, where C = CG(a) is abelian di-group, |a| = ∞, 〈a〉 ⊳ G,
|x| = 2, then G is the group of the type 9) of the theorem 3.
Proof. Since the centre Z(G) is non-periodic and all finite subgroups
of C are invariant in G and Z(G) = 1, then C = 〈b〉λA by the theorem
2.2. from [1], where |b| = pn, p 6= 2, n ≥ 1, A is non-divisible abelian
torsion free subgroup of rank 1 and the commutant C
′
is of prime order
p. If the subgroup C contains all elements of infinite order of the group
G, all p-elements (p 6= 2) and the group G does not contain elements of
order 4, then (xb)2 = 1. Hence x−1bx = b−1.
Let us show, that |b| = p. Let |b| = p2 > p. Then bpn−1
= b1 ∈ Z(G).
Then 〈b1, y〉 ⊳ G for an arbitrary element y ∈ G of infinite order by the
lemma 1. Hence 〈yp〉 ⊳ G and that is why x−1yx = ykbm
1
, x−1ypx =
ykp = y−p. Consequensly, k = −1. Further we have y = x−2yx2 =
x−1(y−1bm
1
)x = (x−1yx)−1 · (x−1b1x)m = yb−m
1
· b−m
1
= yb−2m
1
.
Since p 6= 2, then bm
1
= 1, x−1yx = y−1. Let us take an arbitrary
element a ∈ G. Then |ba| = ∞ and according to the proved x−1(ba)x =
(ba)−1 = a−1b−1 and x−1(ba)x = (x−1bx) · (x−1ax) = b−1a−1. Hence
[a, b] = 1. We obtain the contradiction according to the condition of the
lemma. Thus, |b| = p 6= 2.
The group G contains invariant decomposable abelian subgroups by
66 On groups with the minimal condition
the theorem 2.2. from [1]. Such group can be only the group of order 2p
by the lemma 1. That is why [x, b] = 1.
Let CA(b) = A1 6= A. Then A/A1 is finite cyclic group and that is
why A = 〈a〉A1 for some element a ∈ G.
Let us investigate the possible order of the element xa. Since the
subgroup C contains all elements of infinite order and all p-elements, then
either |xa| = 2p or |xa| = 2. If |xa| = 2p, then
∣
∣(xa)2
∣
∣ = p and (xa)2 = bk,
(k, p) = 1. Hence x−1ax = bka−1. Then a = x−2ax2 = x−1(bka−1)x =
bk · (x−1ax)−1 = bkab−k and that is why a ∈ CA(b). Hence we obtain the
contradiction according to the condition of the lemma. Thus, |xa| = 2
and x−1ax = a−1. Hence x−1yx = y−1 for an arbitrary element y ∈ A.
Let C1 = CG(b). It is evident, that C1 = 〈b, A1, x〉 ⊳ G. Then
[a, x] = a−2 ∈ C1. That is why a−1ba = b−1. At last we have G =
(〈b〉λ 〈a〉A1)λ 〈x〉, |b| = p 6= 2, |x| = 2, 〈a〉A1 = A is non-divisible
abelian torsion free group of rank 1, a2 ∈ A1, [A1, b] = [x, b] = 1,
a−1ba = b−1, x−1yx = y−1 for an arbitrary element y ∈ A. Thus,
the group G is the group of the type 9) of the theorem 3. This completes
the proof of the lemma. This also completes the proof of the theorem.
Corollary. A non-periodic md-group G, which contains decomposable
abelian subgroups, contains non-invariant decomposable abelian subgroups
if and only if, it is the group of the type 5),7), 8), 9) of the theorem 3
or the group of the type 6) of the theorem 3, if it contains the Klein’s
subgroup.
References
[1] Lyman F.M. Groups, in which all decomposable subgroups are invariant // Ukr.
Math.J. - 1970. - V.22, N.6. - P.725-733.
[2] Chernikov S.N. Groups with defined properties of the system of subgroups. -
Moscow: Nauka, 1980. - 384p.
[3] Lyman F.M. Non-periodic groups, in which all decomposable pd-subgroups are
invariant // Ukr. Math.J. - 1988. - V.40, N.3. - P.330-335.
Contact information
F. N. Lyman Full postal address of the first author
E-Mail: f.author@domain.com
M. G. Drushlyak Full postal address of the second author
E-Mail: s.author@domain.com
URL: www.sauthor.domain.com
Received by the editors: 11.08.2006
and in final form 29.03.2006.
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