From: Joachim Neubueser
Resent-Date: Thu, 20 Jul 95 9:22:48 BST
Dear Colleagues,
Martin Isaacs asked in the group-pub-forum:
> Inspired by Problem 4 and some of the commentary on it, I ask
> if any of the GAP or Magma people can throw any light on how
> often character tables can distinguish otherwise very similar
> groups. Specifically, I wonder how many different character
> tables there are for the tens of thousands of different groups
> of order 2^8. What is the average number of different groups
> for a given table; what is the maximum number? I would also
> be interested in the answer when the character tables are
> augmented by the power maps.
The question for Brauer pairs of groups of 2-power order and more
generally for similarities of such groups with respect to character
tables and tables of marks had been frequently asked by John McKay. In
1992 Ellen Skrzipczyk, a student of Herbert Pahlings, in a diploma
thesis, written at Lehrstuhl D, RWTH Aachen, has systematically
investigated Eamonn O'Brien's tables of 2-groups in this respect,
making extensive use of GAP.
She has extensive statistics, and in particular she found Brauer pairs
of groups of order 256, i.e. pairs of nonisomorphic groups, for which
not only the character tables coincide (that after all happens already
for the two nonabelian groups of order 8) but also all information
contained in the table head, in particular the power maps. Moreover,
if I read her report correctly for one of the pairs also the table of
marks coincide.
I say cautiously 'if I read her report correctly', since this thesis
was not supervised by me and I only glanced through a copy this
morning. Prof. Pahlings at present is on holidays and will return only
in mid-August. I will show him Martin Isaacs' request and I expect
that he will come back to the questions posed in more detail and with
more competence then, at present I just wanted to inform that indeed a
study of this kind has been made.
Joachim Neubueser
++++++++++++++++++++++++++++++++++++++++++++++++++
From: David Sibley
Resent-Date: Fri, 21 Jul 95 15:40:55 BST
Marty Isaacs wrote:
>Inspired by Problem 4 and some of the commentary on it, I ask
>if any of the GAP or Magma people can throw any light on how
>often character tables can distinguish otherwise very similar
>groups. Specifically, I wonder how many different character
>tables there are for the tens of thousands of different groups
>of order 2^8. What is the average number of different groups
>for a given table; what is the maximum number? I would also
>be interested in the answer when the character tables are
>augmented by the power maps.
The ten Brauer pairs of order 256 are, by their GAP TwoGroup indices,
1734, 1735
1736, 1737
1739, 1740
1741, 1742
3378, 3380
3379, 3381
3678, 3679
4154, 4157
4155, 4158
4156, 4159
That is, TwoGroup(256,1734) and TwoGroup(256,1735) form the first pair.
This information is (indirectly) from the Thesis previously mentioned
in this mailing list.
David Sibley