For each x S, G x is a maximal (proper) subgroup of G. The proof of

For each x ϵ S, G_x is a maximal (proper) subgroup of G. The proof of this fact proceeds in several steps:

(a) A block of G is a subset T of S such that for each g ϵ G either gT ∩ T = Ø or gT = T, where gT = {gx | x ϵ T}. Show that if T is a block, then |T| divides 7.

(b) If G_x is not maximal, then there is a block T of G such that |T|, ł , contradicting part (a).