On a subalgebra of the centre of a group ring II

Abstract

Let p be a prime, G a finite group with p | |G| and F a field of characteristic p. By $$Z^G_{p^\prime}$$ we denote the F-subspace of the centre of the group ring FG spanned by the p-regular conjugacy class sums. J. Murray proved that $$Z^G_{p^\prime}$$ is an algebra, if G is a symmetric or alternating group. This can be used for the computation of the block idempotents of FG. We proved that $$Z^G_{p^\prime}$$ is an algebra if the Sylow-p-subgroups of G are abelian. Recently, Y. Fan and B. Kulshammer generalized this result to blocks with abelian defect groups. Here, we show that $$Z^G_{2^\prime}$$ is an algebra if the Sylow-2-subgroups of G are dihedral. Therefore $$Z^{PSL(2,q)}_{p^\prime}$$ and $$Z^{PGL(2,q)}_{p^\prime}$$ are algebras for all primes p and all prime powers q. Furthermore we prove that $$Z^{Sz(q)}_{p^\prime}$$ is an algebra for the simple Suzuki-groups Sz(q), where q is a certain power of 2 and p is an arbitrary prime dividing |Sz(q)|.