# Full invariance does not satisfy intermediate subgroup condition

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup)notsatisfying a subgroup metaproperty (i.e., intermediate subgroup condition).

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## Contents

## Statement

### Statement with symbols

It is possible to have groups such that is a fully invariant subgroup of but is not a fully invariant subgroup of .

## Related facts

### Intermediate subgroup condition

- Homomorph-containment satisfies intermediate subgroup condition
- Homomorph-containing implies intermediately fully invariant
- Characteristicity does not satisfy intermediate subgroup condition

## Proof

### Abelian example of prime-cube order

Let be any prime. Consider the group:

.

Let be the set of powers in and be the set of elements of order or . Then , and:

- is fully invariant in : This is on account of it being an agemo subgroup -- the image of a power under an endomorphism continues to be a power.
- is not fully invariant in : In fact, is where , so is an elementary abelian group of order . In particular, has an automorphism interchanging the coordinates, and is not invariant under this automorphism.

### Example of the dihedral group

`Further information: dihedral group:D8, subgroup structure of dihedral group:D8`

Let be the dihedral group of order eight:

.

Let and . Then:

- is fully invariant in : In fact, , and also is the commutator subgroup of , so is fully invariant in .
- is not fully invariant in : In fact, where , so is a Klein four-group and it admits an automorphism interchanging and .