Sylow's Third Theorem: For each prime p dividing the order of a finite group G, the number of Sylow p-subgroups of G is congruent to 1 modulo p and divides |G|. we use the following facts here. 1. By Sylow's theorem, number of 7 sylow subgroups must be 1 or 8(Prove!) 2. number of 7 sylow subgroups must be 8, can not be 1. (Use simplicity of the group) The problem can also be improved as follows.
