Re isomorphic 2-groups of order 1024.

In work on finite groups in general the following will often distinguish
between two non-isomorphic groups of the same order. Compute the order
structure of each group, i.e., the number of elements of order 2, 4 8, etc
and the number of conjugacy classes i.e., you may have 17 elements of order
2 distributed in 5 classes ..etc for order 4, 8 etc, e.g., order 32 there
are two groups of order 32 with 19 elements of order 2 and 12 elements of
order 4, but their distribution into classes is different.(in one case the
12 elements of order 4 are distributed in three classes (4 elements each) in
the other case into 6 classes (2 elements each) In other cases where the
number of elements of a given order is the same, and they are distributed in
the same total number of classes their distribution into the various
classes may be different, this occurs in order 162 (NOT a p-group) but this
probably occurs in p-groups as well.

Another item that often will distinguish different groups is their
automorphism groups. For 2-groups the aut(g) is often another 2-group, but
it may be of a different order, so that will also work.

These are two among many other simple computations that might help,to
distinguish these two groups.

Hope this is useful Walter Becker

From: Alexander B. Konovalov <alexk@mcs.st-and.ac.uk>
Reply-To: GAP Forum <GAP-Forum-Reply@dcs.st-and.ac.uk>
To: Multiple recipients of list <GAP-Forum@dcs.st-and.ac.uk>
Subject: IsomorphismGroups
Date: Sun, 13 Oct 2002 10:22:50 +0100

Dear Forum,

I have a pair of 2-groups of size 1024 which are likely to be
non-isomorhic. On current stage I am interested in obtaining an answer
yes or not, and not in finding an isomorhism if the answer is "yes".
This groups are PcGroups, so they have 10 generators, which is not the
MinimalGeneratingSet. Is it better to generate a pair of another groups
with Group(MinimalGeneratingSet(G)) and then apply IsomorphismGroups to
this new pair ? Are there any another approaches ?

Sincerely yours,
Alexander Konovalov

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