MMN-4944
Monte Carlo method and group algebras
Zsolt Balogh;Abstract
Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of characteristic $p$.
Let $\cd$ be an involution of $FG$ and $V_{\cd}(FG)$ the $\cd$-unitary subgroup of $FG$.
The order of $V_{\cd}(FG)$ is known when $p$ is an odd prime, and $\cd$ arises from $G$, however the case of two characteristic is a challenging problem.
The RAMEGA package of GAP system contains implementations of functions based on random methods related to group algebras. In this paper we provide the theoretical background of some random functions of RAMEGA that related to the $*$-unitary subgroup of $FG$. Furthermore, for the canonical involution $*$ we estimate the order of the $*$-unitary subgroup of $FG$ for non-abelian $2$-groups of order $2^5$
using Monte Carlo method.
Vol. 26 (2025), No. 1, pp. 61-67