# Abelian subnormal subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: subnormal subgroup with a group property (itself viewed as a subgroup property): abelian group

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed an **abelian subnormal subgroup** if is an abelian group as a group in its own right (or equivalently, an abelian subgroup of ) and is a subnormal subgroup of .

## Examples

Here are some examples of subgroups in basic/important groups satisfying the property:

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

A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |

Z2 in V4 | Klein four-group | Cyclic group:Z2 | Cyclic group:Z2 |

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

abelian 2-subnormal subgroup | |FULL LIST, MORE INFO | |||

abelian normal subgroup | abelian and a normal subgroup -- invariant under all inner automorphisms | from normal implies subnormal | |FULL LIST, MORE INFO | |

abelian characteristic subgroup | abelian and a characteristic subgroup -- invariant under all automorphisms | (via normal) | |FULL LIST, MORE INFO | |

abelian fully invariant subgroup | abelian and a fully invariant subgroup -- invariant under all endomorphisms | Abelian characteristic subgroup|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

nilpotent subnormal subgroup | |FULL LIST, MORE INFO |