On fixed point sets of distinguished collections for groups of parabolic characteristic

On the vertices of indecomposable summands of certain Lefschetz modules

“…It asserts that under certain assumptions on the shape of the centralizer of a purely noncentral p-subgroup T of G, the structure of the fixed point set D.G/ T is determined by the collection of p-centric and p-radical subgroups in a subquotient (or section) of this centralizer. Our Theorem 3.6 generalizes the result of [29,Theorem 4.14]. Before stating and proving the theorem, we need to introduce some more notation.…”

“…The next result was first obtained for T a cyclic group of order p in [29], and the arguments from Propositions 4:4, 4:5 and 4:7 of that paper also apply to any p-group T . For completeness we include these proofs below.…”

“…The proof in [29] of this homotopy equivalence can actually be seen to yield equivariance under any automorphism of H , so that here we have equivariance under N G .T /=O C , or under N G .T / with O C acting trivially.…”

“…For a group G of parabolic characteristic p, the complex D p .G/ of p-centric and p-radical subgroups equals the complex b B p .G/ of distinguished p-radical subgroups. If the largest normal p-subgroup O p .T C G .T // of T times its centralizer contains p-central elements then the fixed point set D p .G/ T is contractible; this is Theorem 3.3 and was initially proven in [29] for T cyclic. If O p .T C G .T // is purely noncentral, we assume that T C G .T / has a component with central quotient H which also has parabolic characteristic p. This assumption seems natural if we keep in mind the p-local structure of the sporadic simple groups of parabolic characteristic p. Our main result Theorem 3.6 asserts that the fixed point set D p .G/ T is equivariantly homotopy equivalent to the complex of distinguished p-radical subgroups in H .…”

“…(ii) [29,Propostion 3.6] If G has local characteristic p, then 2.13. In the case that G has parabolic characteristic p, the better known collection of p-centric p-radical subgroups is equal to the collection of distinguished p-radical subgroups.…”