Let p be the smallest prime dividing order of a finite group G,show any subgroup H of G of index p is normalgo

Question

Let p be the smallest prime dividing order of a finite group G,show any subgroup H of G of index p is normalgroup

Answer 1

Left multiplication of the cosets of H by an element of G permutes the cosets, resulting in a homomorphism from G into S(p), the permutation group on p elements.

Let K be the image of G in S(p) under this homomorphism. Then:

The order of K divides the order of S(p)

The order of S(p) is p!

The order of G is the product of the orders of the kernel of the homomorphism and of K

The only prime shared by the order of G and p! is p, which occurs only once in p!

Therefore the order of k is p, the order of the kernel is the quotient of the order of G by p, which is also the order of H. So H is a subgroup of the kernel of the same cardinality as the kernel, so H is the kernel of the homomorphism.

Kernels of group homomorphisms are always normal subgroups

So H is a normal subgroup of G