# Subgroup-defining function value is characteristic

From Groupprops

## Contents

## Definition

Suppose is a subgroup-defining function, i.e., associates to every group a subgroup , with the property that given an isomorphism , the image of under is .

Then, for every group , is a characteristic subgroup of .

## Related facts

### Conclusion about normality

Combined with the fact that characteristic implies normal, this tells us that any subgroup-defining function value is normal.

### Applications to specific subgroup-defining functions

Subgroup-defining function | Meaning | Proof that it is characteristic | Proof of stronger properties satisfied for the particular subgroup-defining function |
---|---|---|---|

center | elements that commute with everything | center is characteristic | center is bound-word, center is strictly characteristic, center is direct power-closed characteristic |

derived subgroup | products of commutators | derived subgroup is characteristic | derived subgroup is verbal, derived subgroup is fully invariant |

Frattini subgroup | intersection of all maximal subgroups | Frattini subgroup is characteristic |

## Proof

**Given**: A group , a subgroup-defining function , an automorphism of .

**To prove**: .

**Proof**: Since is an automorphism, it is in particular an isomorphism. Thus, the definition of subgroup-defining function above, setting , gives us what we need to prove.