> < ^ Date: Wed, 16 Dec 1992 19:01:33 +0100
> ^ From: R. Keith Dennis <dennis@rkd.math.cornell.edu >
^ Subject: commutator questions

In case this is not the right place to ask this question, let me ask
another: Is there a group theory discussion list on the net? If not,
should there be one?

Commutator Properties of Groups

The properties defined below arose several years ago in work with
R. Alperin in computing the second homology group H_2(G,Z).

A group G will be said to satisfy the property C_n if for every
collection of n elements x_1,...,x_n in G , there exist elements
z, y_1,...,y_n in G so that for each i = 1,...,n, x_i = [z,y_i].

A group G will be said to satisfy the property C'_n if for every
collection of n elements x_1,...,x_n in G' , there exist elements
z, y_1,...,y_n in G so that for each i = 1,...,n, x_i = [z,y_i].

Thus a group satisfies C_1 precisely when every element of G is
a commutator. It is well-known that for n >= 5, A_n satisfies C_1.
Several years ago, Ulf Rehmann (Bielefeld) and I checked that A_n
satisfies C_2 for 5 <= n <= 10. As I recall, A_5 did not
satisfy C_3. If U is the multiplicative group of real quaternions
of norm 1, then one can show that U satisfies C_3.

One might reasonably conjecture that A_n satisfies C_2 for n >= 5.
One might also ask if there is a function c(n) so that A_m satisfies
C_n provided that m >= c(n). One would then presumably have
c(1) = 5, c(2) = 5, and c(3) > 5. I have no evidence that such
a function should exist.

What results of this type are known for other groups, for example,
PSL_n(F) or the non-abelian finite simple groups? How easy is it to
use GAP to study questions of this type? How large can they be? The
computations mentioned earlier used a special purpose C program and
took lots of memory and time.

Keith Dennis

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