# Characteristicity is not finite direct power-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup)notsatisfying a subgroup metaproperty (i.e., finite direct power-closed subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about characteristic subgroup|Get more facts about finite direct power-closed subgroup property|

## Contents

## Statement

### Statement with symbols

It is possible to have a group and a characteristic subgroup of such that in the direct square , the corresponding direct square , viewed as a subgroup, is *not* a characteristic subgroup.

## Related facts

- Full invariance is finite direct power-closed, hence, fully invariant implies finite direct power-closed characteristic

## Notion of finite direct power-closed characteristic

A subgroup of a group is termed a finite direct power-closed characteristic subgroup if for any natural number , is characteristic in .

## Proof

`Further information: direct product of Z8 and Z2`

Suppose , and is the subgroup of order four given by:

Note that this subgroup is characteristic but not fully invariant, and is a standard example of the fact that characteristic not implies fully invariant in finite abelian group.

Then, we claim that is *not* a characteristic subgroup inside . To see this, consider the endomorphism of that projects onto the coordinate, i.e., the map:

Now, consider the automorphism of defined as:

This is an endomorphism as it is a sum of endomorphisms; it is an automorphism because it has an inverse:

In full, the automorphism is:

Under this automorphism, the element:

gets mapped to the element:

which is *not* inside .