Fixed point free elements of prime order - School of Maths and Stats ...

FKS examples againAGL(1, p 2 ) ∼ = C 2 p ⋊ C p 2 −1Stabiliser of a line is isomorphic to C p ⋊ C p−1G 1 = C p 2 −1 GL(2, p), W a 1-dimensional subspace, H 1 = C p−11 G 1 acts transitively on nontrivial elements of V .2 Only primes dividing p 2 − 1 are those dividing p − 1.In fact, any group C 2 p ⋊ G 1 , where G 1 C p 2 −1 contains theSylow 2-subgroup, is elusive.