Thread: cauchy theorem

If p is a prime and G is a finite group in which every element has order a power of p, prove that G is a p-group.


I was think cauchy theorem would help here.... If G is a finite group whose order is divisible by a prime p, the G contains an element of order p.

Originally Posted by ux0

If p is a prime and G is a finite group in which every element has order a power of p, prove that G is a p-group.


I was think cauchy theorem would help here.... If G is a finite group whose order is divisible by a prime p, the G contains an element of order p.

Of course it helps...and a lot: if G is not a p-group then there's some prime q different from p that divides the order of G ==> by Cauchy's theorem G has an element of order q, contradiction.

Tonio