On stable finiteness of group rings
For an arbitrary field or division ring K and an arbitrary group G, stable finiteness of K[G] is equivalent to direct finiteness of K[G×H] for all finite groups H.
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| Cite this: | On stable finiteness of group rings / K. Dykema, K. Juschenko // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 44-47. — Бібліогр.: 4 назв. — англ. |
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Dykema, K. Juschenko, K. 2019-06-12T20:51:05Z 2019-06-12T20:51:05Z 2015 On stable finiteness of group rings / K. Dykema, K. Juschenko // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 44-47. — Бібліогр.: 4 назв. — англ. 1726-3255 2000 MSC:20C07. https://nasplib.isofts.kiev.ua/handle/123456789/152785 For an arbitrary field or division ring K and an arbitrary group G, stable finiteness of K[G] is equivalent to direct finiteness of K[G×H] for all finite groups H. Research supported in part by NSF grants DMS–0901220 and DMS-1202660. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On stable finiteness of group rings Article published earlier |
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On stable finiteness of group rings |
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On stable finiteness of group rings Dykema, K. Juschenko, K. |
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on stable finiteness of group rings |
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Algebra and Discrete Mathematics |
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For an arbitrary field or division ring K and an arbitrary group G, stable finiteness of K[G] is equivalent to direct finiteness of K[G×H] for all finite groups H.
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On stable finiteness of group rings / K. Dykema, K. Juschenko // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 1. — С. 44-47. — Бібліогр.: 4 назв. — англ. |
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2025-11-25T20:31:26Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 19 (2015). Number 1, pp. 44–47
© Journal “Algebra and Discrete Mathematics”
On stable finiteness of group rings
Ken Dykema1, Kate Juschenko
Communicated by R. I. Grigorchuk
Abstract. For an arbitrary field or division ring K and an
arbitrary group G, stable finiteness of K[G] is equivalent to direct
finiteness of K[G × H] for all finite groups H.
1. Introduction
Kaplansky’s Direct Finiteness Conjecture (DFC) asks whether for
every field (or division ring) K and every group G, the group ring K[G]
is directly finite, namely, whether a, b ∈ K[G] and ab = 1 implies ba = 1
(where 1 denotes the identity element of K[G]. Kaplansky showed that
the answer is positive when K has characteristic 0 — see [4], p. 122. One
can also ask for more: one can ask for all matrix algebras Mn(K[G]) over
the group ring to be directly finite. If this holds, the group ring K[G]
is said to be stably finite. Stable finiteness of K[G] was shown (at the
same time as direct finiteness) by Ara, O’Meara and Perera [1] when G is
residually amenable and by Elek and Szabó [3] when G is (more generally)
sofic. In this note, we show that stable finiteness of K[G] is equivalent to
direct finiteness of K[G × H] for all finite groups H.
Though the argument is not difficult, we believe the result is of interest
because it may offer a short-cut for proving stable finiteness for certain
classes of groups. Our investigation was motivated by the approach of [2]
1Research supported in part by NSF grants DMS–0901220 and DMS-1202660.
2000 MSC: 20C07.
Key words and phrases: Kaplansky’s Direct Finiteness Conjecture, stable finite-
ness.
K. Dykema, K. Juschenko 45
to the testing the DFC by testing certain finitely presented groups – the
Universal Left Invertible Element, or ULIE, groups. At first we wondered
whether an analogous effort should be made for stable finiteness, but
Corollary 2.3 shows that there would be no advantage in doing so.
2. Stable finiteness
Lemma 2.1. Given a field F and a positive integer n, there is a fi-
nite group H such that the group ring F [H] has a nonunital subalgebra
isomorphic to Mn(F ).
Proof. We prove first the case n = 2. Let p be the characteristic of the field
F . Consider the symmetric group S3 = 〈a, b | a3 = b2 = 1, bab = a−1〉.
Consider the representation π of S3 on F 2 given by
π(a) =
(
−1 −1
1 0
)
, π(b) =
(
0 1
1 0
)
,
extended by linearity to a representation of F [S3]. We have
π(a2) =
(
0 1
−1 −1
)
, π(ab) =
(
−1 −1
0 1
)
, π(a2b) =
(
1 0
−1 −1
)
.
An easy row reduction computation shows that when p 6= 3, we have
span π(S3) = M2(F ). Let us assume p 6= 3. In the case p > 3, the desired
conclusion of the lemma will follow from Maschke’s theorem, but by
performing the actual computations, we will now see that the conclusion
holds also for p = 2. Let Q = 1
3
(2 − a − a2) ∈ F [S3]. Then Q2 = Q, and
Q(F [S3])Q is a subalgebra of F [S3]. An easy computation shows that
Q(F [S3])Q has dimension 4 over F , and π(Q) = ( 1 0
0 1
); this implies that
the restriction of π to Q(F [S3])Q is an isomorphism onto M2(F ) and
Q(F [S3])Q ∼= M2(F ) as algebras. The lemma is proved in the case of
n = 2 and p 6= 3.
We now suppose p > 2 and consider the dihedral group of order 8
Dih4 = 〈c, d | c4 = d2 = 1, dcd = c−1〉
and its representation on F 2 given by
σ(c) =
(
0 −1
1 0
)
, σ(d) =
(
1 0
0 −1
)
,
46 On stable finiteness of group rings
which gives
σ(c2) =
(
−1 0
0 −1
)
, σ(c3) =
(
0 1
−1 0
)
,
σ(cd) =
(
0 1
1 0
)
, σ(c2d) =
(
−1 0
0 1
)
, σ(c3d) =
(
0 −1
−1 0
)
.
We easily see span σ(Dih4) = M2(F ). Now the result follows by Maschke’s
theorem, but let us perform the easy calculation. Letting Q = 1
2
(1 − c2) ∈
F [Dih4], we have Q2 = Q and Q(F [Dih4])Q is a subalgebra of F [Dih4].
We have σ(Q) = ( 1 0
0 1
) and dim(Q(F [Dih4])Q) = 4 and the restriction of
σ to Q(F [Dih4])Q is an isomorphism onto M2(F ). Thus, the lemma is
proved in the case n = 2 and p > 2. Taken together, these considerations
prove the lemma in the case of n = 2.
For groups H1 and H2 we have the natural identification F [H1 ×H2] ∼=
F [H1] ⊗F F [H2], and for positive integers m and n we have Mm(F ) ⊗F
Mn(F ) ∼= Mmn(F ). Therefore, starting from the case n = 2 of the
lemma and taking cartesian products of an appropriate group, arguing
by induction we prove the lemma in the case when n is a power of 2.
Now taking corners of the matrix algebras M2k(F ) proves the lemma for
arbitrary n.
Theorem 2.2. Let K be a field or a division ring and let Γ be a group.
Then the following are equivalent:
(a) K[Γ] is stably finite
(b) K[Γ × H] is directly finite for every finite group H.
Proof. Suppose (a) holds and let H be a finite group. Let F be the base
field of K. Then H acting on itself by permutations yields an injective,
unital algebra homomorphism F [H] → Mn(F ), where n is the order of H,
and we may regard F [H] as a unital subalgebra of Mn(F ). Then
K[Γ × H] ∼= K[Γ] ⊗F F [H] ⊆ K[Γ] ⊗F Mn(F ) ∼= Mn(K[Γ]),
with the inclusion being unital, and direct finiteness of K[Γ × H] follows
from that of Mn(K[Γ]).
Suppose (b) holds. Let n be a positive integer. By Lemma 2.1, choose
a finite group H so that F [H] contains Mn(F ) as a nonunital subalgebra.
Then F [H] contains a copy of Mn(F ) ⊕ F as a unital subalgebra, and
we have
K[Γ × H] ∼= K[Γ] ⊗F F [H] ⊇ K[Γ] ⊗F (Mn(F ) ⊕ F )
∼= (K[Γ] ⊗F Mn(F )) ⊕ K[Γ] ∼= Mn(K[Γ]) ⊕ K[Γ] ⊇ Mn(K[Γ]) ⊕ K,
K. Dykema, K. Juschenko 47
where all inclusions are as unital subalgebras. Now given c, d ∈ Mn(K[Γ])
such that cd = 1, take a = c ⊕ 1 and b = d ⊕ 1 in Mn(K[Γ]) ⊕ K. We have
ab = 1, and by the above inclusions and the direct finiteness of K[Γ × H],
we must have ba = 1, so dc = 1.
Consequently, truth of the Direct Finiteness Conjecture implies truth
of the stronger looking Stable Direct Finiteness Conjecture.
Corollary 2.3. For K is a division ring, if K[Γ] is directly finite for all
groups Γ, then K[Γ] is stably finite for all groups Γ.
Remark 2.4. From the proofs of Lemma 2.1 and Theorem 2.2, we see
that the conditions (a) and (b) of Theorem 2.2 are also equivalent to the
following:
(c) Let H1 = Dih4 if the characteristic of K is 3 and otherwise let
H1 = S3. Then K[Γ×H] is directly finite whenever H is a Cartesian
product of finitely many copies of H1.
References
[1] P. Ara, K. O’Meara, and F. Perera, Stable finiteness of group rings in arbitrary
characteristic, Adv. Math. 170 (2002), 224–238.
[2] K. Dykema, T. Heister, and K. Juschenko, Finitely presented groups related to
Kaplansky’s Direct Finiteness Conjecture, Exp. Math., to appear.
[3] G. Elek and E. Szabó, Sofic groups and direct finiteness, J. Algebra 280 (2004),
426–434.
[4] I. Kaplansky, Fields and rings, The University of Chicago Press, 1969.
Contact information
Ken Dykema Department of Mathematics, Texas A&M Uni-
versity, College Station, TX 77843-3368, USA
E-Mail(s): kdykema@math.tamu.edu
Kate Juschenko Department of Mathematics, Northwestern Uni-
versity, 2033 Sheridan Road, Evanston, IL 60208-
2730, USA
E-Mail(s): kate.juschenko@gmail.com
Received by the editors: 06.12.2014
and in final form 06.12.2014.
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