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"Let G be a finite group. Prove that G is a p-group iff every element of G has order a power of p."

this was my attempt at the question:

Suppose G is a p-group

Let g

Hence, every element of G has order a power of p.

Conversely, suppose that every element of G has order a power of p

But

Hence, G is a p-group.

can someone look over this and tell me what mistakes i made if any please ?