Center not is fully invariant

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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., fully invariant 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 a fully invariant subgroup.

Related facts

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.

Examples

 Group partSubgroup partQuotient part
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group