[SaSA :::::]

There are some unnecessary arguments in my previous attachment, please refer this, it's better.

[SaSA :::::]

I hadn't read about direct products up to this problem so it didn't come to me to use them, and I completely missed that solution of part two could be used. Thanks a lot for your solution, using it I was able to find an alternative which avoids direct products, essentially the idea is to show that the center of P is in the center of G, which is also what your solution does, thanks again!

[SaSA :::::]

Thanks, I saw this, but the problem is that Herstein has not introduced much of these ideas up to this problem. I found something different, it goes like this, center Z of G has order either 1 or pq, if q doesn't divide p-1 it can be shown that a normal subgroup of order p is contained in Z, therefore Z=G, so G is abelian and it has an element of order pq.

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