Characteristic not implies fully invariant in odd-order class two p-group

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a odd-order class two p-group. That is, it states that in a odd-order class two p-group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., fully invariant subgroup)
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Statement

For any (odd) prime p, there exists a p-group G of class two and a characteristic subgroup of this group that is not fully invariant.

The construction also works for p = 2, but for p = 2, there are already examples of abelian groups with characteristic subgroups that are not fully invariant.

Related facts

Proof

Let p be an odd prime. Let P be any non-abelian group of order p^3 with center Z. There are two possibilities for P: a group of prime-square exponent, and a group of prime exponent. In both such groups, there is an element x of order p outside Z.

Define G = P \times C where C is the cyclic group of order p with generator y. The center of G is the subgroup H = Z \times C. Then:

  • H is characteristic in G, because center is characteristic.
  • H is not fully invariant in G: Consider the retraction with kernel P \times \{ e \} and with image generated by the element (x,y). This is an endomorphism of G, but it does not send H to itself, since the element (e,y) gets sent to (x,y), which is outside H.