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