# Solvable radical is normal-homomorph-containing

From Groupprops

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., solvable radical) always satisfies a particular subgroup property (i.e., normal-homomorph-containing subgroup)}

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Contents

## Statement

Suppose is the solvable radical of a group , i.e., the *unique* largest solvable normal subgroup of . (Note: The solvable radical may not exist for every group. It does, however, exist for finite groups, virtually solvable groups, and slender groups).

Then, is a normal-homomorph-containing subgroup of . In other words, for any homomorphism such that is a normal subgroup of , .

## Related facts

### Stronger facts: stronger subgroup properties satisfied

Property | Meaning | Proof of satisfaction by solvable radical | Why it's stronger than normal-homomorph-containing |
---|---|---|---|

Normal-subhomomorph-containing subgroup | contains any normal subgroup that is a homomorphic image of a subgroup | solvable radical is normal-subhomomorph-containing | obvious |

### Weaker facts: weaker subgroup properties satisfied

Property | Meaning | Proof of satisfaction by solvable radical | Why it's weaker than normal-homomorph-containing |
---|---|---|---|

weakly normal-homomorph-containing subgroup | invariant under homomorphisms from it to whole group that send normal subgroups to normal subgroups | normal-homomorph-containing implies homomorph-containing | |

Strictly characteristic subgroup | invariant under all surjective endomorphisms | solvable radical is strictly characteristic | normal-homomorph-containing implies strictly characteristic |

Characteristic subgroup | invariant under all automorphisms | solvable radical is characteristic | (via strictly characteristic) |

Normal subgroup | invariant under all inner automorphisms | solvable radical is normal | (via characteristic) |