# Group satisfying Oliver's condition

## Definition

A group satisfying Oliver's condition is a finite group $G$ satisfying the following equivalent conditions:

• $G$ is a $q$-group extension of a cyclic extension of a $p$-group: There exist primes $p,q$ (possibly equal) and subgroups $N \le H \le G$, such that the following hold. $N$ is a normal subgroup of $H$, $H$ is a normal subgroup of $G$, $G/H$ is a $q$-group, $H/N$ is a cyclic group, and $N$ is a $p$-group. By a $p$-group (resp., $q$-group), we mean a group of prime power order where the underlying prime is $p$ (resp., $q$).
• Any action of $G$ on a contractible finite simplicial complex has a fixed face (in the geometric realization, a fixed point).