[GAP Forum] Extraspecial groups and orthogonal over Z2

mim_ at op.pl mim_ at op.pl
Wed Jan 6 15:01:02 GMT 2010


Please forgive me sending this post on GAP forum. I don't know any forum for finite groups. If you know such then please let me know. I tried to register on group-pub-forum at maths.bath.ac.uk but it didn't work.

I have discovered lately that extraspecial groups 2^(2n+1) can be represented as monomials in Clifford algebra C(2n). There are two such algebras depending on how we choose generators to square to +1 or to -1. 2^2n choices but in effect only two algebras are obtained. E.g. for C4 we have algebra C4=M2H obtained for ++++, ----, ---+. Algebra M4R=H*H is obtained for +-+-, +++- choices. I don't know yet the general principle for this but this must be easy.

Conway paper "Simple construction of monster" (1984) in appendix says that automorphism of extraspecial group is equal to O+(2n,2) or O-(2n,2) times 2^2n. I have checked for n=2 that this is really true. This is nice way to define O+ and O- over Z2.

Can anybody share thoughts on this subject ? Any literature I can find about it ?

Marek Mitros

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