MONTE CARLO METHOD AND GROUP ALGEBRAS

Let FG be the group algebra of a finite p-group G over a finite field F of characteristic p. Let ⊛ be an involution of FG and V⊛(FG) the ⊛-unitary subgroup of FG. The order of V⊛(FG) is known when p is an odd prime, and ⊛ 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, where ∗ is the canonical involution. We estimate the order of the ∗-unitary subgroup of FG for non-abelian 2-groups of order 2⁵ using Monte Carlo method. Furthermore, we verify the estimated orders for certain groups of order 2⁵.