Center not is fully invariant

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., fully invariant subgroup)
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Statement

The center of a group need not be a fully invariant subgroup.

Related facts

• Center not is fully invariant in class two p-group
• Center not is normality-preserving endomorphism-invariant: We can use the same idea, but need to impose some more constraints on the generic example.

Proof

Generic example

Let $A$ be an abelian group and $C$ a centerless group containing a subgroup isomorphic to $A$, say $B$. Consider the direct product $A\times C$.

Clearly, $A$ is the center of $A\times C$.

Now consider the endomorphism of $A\times C$ which composes the projection onto $A$ with the isomorphism from $A$ to $B$. This endomorphism does not send $A$ to within itself, and hence, $A$ is not a fully characteristic subgroup of $C$.

Thus, the center is not fully invariant.

Particular example

The particular example can be obtained from the generic example above by setting $A$ to be a cyclic group of order 2 and $C=S_{3}$ (the symmetric group on three letters). $B$ in this case is the 2-element subgroup generated by a transposition.