关于群的幂自同态的一个注记

A note on power automorphisms of groups

设f:G→G是群G的自同态,满足f(x)=xn(x∈G),证明了G是交换群当且仅当n=-1或2;设M={n|f:G→G是群G的自同态,满足f(x)=xn,x∈G},证明了G是交换群当且仅当n遍历M中所有元时,所有形如n(n-1)元的最大公因数为2.

Let f：G→G be an homomorphism of a group G, satisfying f（x） =xn（？x∈G）, prove that G is commutative if and only if n=-1 or 2.Let M={n｜f：G→G is homomorphism of G, satisfying f（ x） =xn ,？x∈G} prove that G is com-mutative if and only if the greatest common multiples of numbers of the form of n（n-1）, when n is running over the set M, is 2.