Group satisfying Oliver's condition
A group satisfying Oliver's condition is a finite group satisfying the following equivalent conditions:
- is a -group extension of a cyclic extension of a -group: There exist primes (possibly equal) and subgroups , such that the following hold. is a normal subgroup of , is a normal subgroup of , is a -group, is a cyclic group, and is a -group. By a -group (resp., -group), we mean a group of prime power order where the underlying prime is (resp., ).
- Any action of on a contractible finite simplicial complex has a fixed face (in the geometric realization, a fixed point).