The Order of the Unitary Subgroups of Group Algebras

2022/06/05 by Balogh, Zsolt Adam
#FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2206.02113

Abstract

Let FG be the group algebra of a finite p-group G over a finite field F of positive characteristic p. Let \cd be an involution of the algebra FG which is a linear extension of an anti-automorphism of the group G to FG. If p is an odd prime, then the order of the \cd-unitary subgroup of FG is established. For the case p=2 we generalize a result obtained for finite abelian 2-groups. It is proved that the order of the *-unitary subgroup of FG of a non-abelian 2-group is always divisible by a number which depends only on the size of F, the order of G and the number of elements of order two in G. Moreover, we show that the order of the *-unitary subgroup of FG determines the order of the finite p-group G.

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