On a subalgebra of the centre of a group ring II

摘要

Let p be a prime, G a finite group with p | |G| and F a field of characteristic p. By ZG_p¢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 ZG_p¢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 ZG_p¢Z^G_{p^prime} is an algebra if the Sylow-p-subgroups of G are abelian. Recently, Y. Fan and B. Külshammer generalized this result to blocks with abelian defect groups. Here, we show that ZG_2¢Z^G_{2^prime} is an algebra if the Sylow-2-subgroups of G are dihedral. Therefore ZPSL(2,q)_p¢Z^{PSL(2,q)}_{p^prime} and ZPGL(2,q)_p¢Z^{PGL(2,q)}_{p^prime} are algebras for all primes p and all prime powers q. Furthermore we prove that ZSz(q)_p¢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)|.