# Normality-preserving endomorphism-invariant subgroup

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

A subgroup of a group is termed a **normality-preserving endomorphism-invariant subgroup** if, for every normality-preserving endomorphism of , is contained in . A normality-preserving endomorphism is an endomorphism with the property that the image of any normal subgroup is normal.

## Examples

### Extreme examples

- The trivial subgroup is normality-preserving endomorphism-invariant in any group.
- Every group is normality-preserving endomorphism-invariant in itself.

### Examples arising from stronger properties or subgroup-defining functions

- All fully invariant subgroups, including the derived subgroup (commutator subgroup), as well as members of the derived series and lower central series, are normality-preserving endomorphism-invariant.
- The Fitting subgroup and solvable radical are both normality-preserving endomorphism-invariant. In fact, they are something stronger: weakly normal-homomorph-containing subgroups.

### Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property normality-preserving endomorphism-invariant subgroup.

Group part | Subgroup part | Quotient part | |
---|---|---|---|

Cyclic maximal subgroup of dihedral group:D8 | Dihedral group:D8 | Cyclic group:Z4 | Cyclic group:Z2 |

Below are some examples of a proper nontrivial subgroup that *does not* satisfy the property normality-preserving endomorphism-invariant subgroup.

Group part | Subgroup part | Quotient part | |
---|---|---|---|

Diagonally embedded Z4 in direct product of Z8 and Z2 | Direct product of Z8 and Z2 | Cyclic group:Z4 | Cyclic group:Z4 |

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]