Prove that G has a subgroup of order p Let G be a group of o

Prove that G has a subgroup of order p.

Let (G,*) be a group of order p^2, where p is a prime integer. Prove that G has a subgroup of order p.

Solution

Let x be an element of G other than the identity. { if x=e then order =1 , not possible, p cant be one as it is prime)

By Lagrange\'s Theorem, you know that the order of x must divide |G|=p^2, and since the only divisors of p^2 are 1, p, and p^2, that means that |x|=1, p, or p^2.

If it is p^2,then it will create G.

if it is p, order of subgroup =p

1 is not possible. since 1 is not prime.