# 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)
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.

## 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