Group satisfying Oliver's condition

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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).


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