Nontrivial $\mathscr{P}$(G)-matched $\frak{S}$-related pairs for finite gap Oliver groups

Abstract

In this paper we construct nontrivial pairs of $\frak{S}$-related (i.e. Smith equivalent) real G-modules for the group G = PΣL(2,27) and the small groups of order 864 and types 2666, 4666. This and a theorem of K. Pawalowski-R. Solomon together show that Laitinen's conjecture is affirmative for any finite nonsolvable gap group. That is, for a finite nonsolvable gap group G, there exists a nontrivial $\mathscr{P}$(G)-matched pair consisting of $\frak{S}$-related real G-modules if and only if the number of all real conjugacy classes of elements in G not of prime power order is greater than or equal to 2.