# Center not is fully invariant

From Groupprops

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) doesnotalways 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

- 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 be an abelian group and a centerless group containing a subgroup isomorphic to , say . Consider the direct product .

Clearly, is the center of .

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

Thus, the center is not fully invariant.

### Particular example

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