Center not is image-closed characteristic

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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., image-closed characteristic subgroup)
View subgroup property satisfactions for subgroup-defining functions | View subgroup property dissatisfactions for subgroup-defining functions

Statement

The center of a group need not be an image-closed characteristic subgroup, i.e., its image under a surjective homomorphism need not be a characteristic subgroup of the image.

Proof

Generic example

Let A be an Abelian group and G a centerless group such that A is isomorphic to the quotient of G by some normal subgroup N. Consider the direct product A \times G.

Clearly, A is the center of A \times G.

Now consider the quotient map by N. Under this quotient map, the group A \times G maps to A \times G/N, with A mapping to the direct factor A in A \times G/N. By our assumption, G/N is isomorphic to A, so this quotient group looks like A \times A. Clearly, A is not characteristic in this.

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 G = S_3 (the symmetric group on three letters). N in this case is the subgroup of order three generated by a 3-cycle.