Free Actions on Spheres

The problem of determining which finite groups can act freely on a sphere was first addressed by P. A. Smith in 1944, and resolved by I. Madsen, C. B. Thomas and C. T. C. Wall thirty years later. In this essay we show the negative part of the answer, namely that groups $\mathbb{Z}/p \oplus \mathbb{Z}/p$ and dihedral groups $D_p$ cannot act freely on a sphere for any prime $p$. We also summarise T. Petrie’s construction of a free action on $S^{2q−1}$ of a non-abelian group of order $pq$, with $p, q$ odd primes.