00:01
So here we have an if and only if statement.
00:02
We want to prove that g has order p to the n, where p is prime.
00:06
If and only if the order of every element of g is the power of p.
00:12
So first we're going to let g be a finite abelian group.
00:29
So now first we're going to go starting from g has order p to the n.
00:37
So first suppose order of g equals p to the n or some prime p.
01:00
Then the order every element divides p to the n.
01:21
So the order every element must then be a power of p, as those are the only divisors of p to the n.
01:54
So then going to go the other way.
01:58
Now suppose the order of every element of every element is a power.
02:16
Once again p is a prime...