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# The group is abelian

Let be a finite group. If forall there exists such that then prove that is abelian.

Solution

Since each automorphish preserves the elements’ order , then all elements have the same order. This order ought to be a prime number. It is obvious from Cauchy’s theorem that the order of the group is where . For it is well known that the group is abelian. For we work as follows:

Let . Then its centre is not trivial. Hence there exists such that . It suffices to prove that . Let . Hence:

and the last one is true since . Hence and thus the group is abelian and we are done.

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