Let G be a finite group H a subgroup and N a normal subgroup

Let G be a finite group, H a subgroup, and N a normal subgroups of G. If gcd(|H|. [G: N]) = 1. then H N.

Solution

Since H is a subgroup and N a normal subgroup. HN is a subgroup (because HN = NH; let h1*n1, h2*n2 be elements of HN. (h1*n1)*(h2*n2)^-1 is in HN as N is normal, thus proving HN to be a subgroup.). Let a1,a2,a3 ... be natural numbers. Now |HN| = a1 |H| = a2 |N|.

|G| = a3 |N| and |G| = a4 |HN| and |G| = a5 |H| because these are subgroups. gcd(|H|,a3) = 1. Therefore gcd(|H|,a2) = 1, which means a2 = 1 or |HN| = |N| or HN = N or H is a subset of N.