Finite group actions on the 7-sphere (Q1347843)

By only using classical results about finite group actions on spheres, the authors obtain the following two results: (1) Let \(\pi\) be a finite group acting freely and topologically on \(S^7\). Then \(\pi\) can also act freely and linearly on \(S^7\). (2) For odd natural numbers \(2s-1\) different from 1, 3 or 7, one can always find a free smooth action on \(S^{2s-1}\) by a finite group which cannot act freely and linearly.