Fixed‐point posets of groups and Euler characteristics

Suppose that is a group and is a ‐set. For a set of subgroups of , we introduce the fixed‐point poset . A variety of results concerning are proved as, for example, in the case when is a prime and is a non‐empty set of finite non‐trivial ‐subgroups of of bounded order. Under certain conditions we show that the Euler characteristic of is congruent to 1 modulo , where is the largest power of dividing for any . Brown's homological Sylow theorem is a special case of this result. By making various choices for we observe many other results on Euler characteristics, as well as presenting explicit calculations for specific groups.