Center not is fully invariant in class two p-group
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Let be a prime. Let be any non-abelian group of order with center (if , choose to be dihedral group:D8. Otherwise there are two possibilities for : a group of prime-square exponent, and a group of prime exponent). In all these groups, there is an element of order outside .
Define where is the cyclic group of order with generator . The center of is the subgroup .
Then is not fully invariant in : Consider the retraction with kernel and with image generated by the element . This is an endomorphism of , but it does not send to itself, since the element gets sent to , which is outside .