# Abelian marginal subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: marginal 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 marginal subgroup** if is an abelian group (i.e., it is an abelian subgroup of ) and is also a marginal subgroup of .

## Examples

- The center in any group is an abelian marginal subgroup.
- In a finite p-group , the socle coincides with , the set of elements of order dividing in the center (see socle equals Omega-1 of center in nilpotent p-group). This is an abelian marginal subgroup.

### Examples in small finite groups

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

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

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

Center of dihedral group:D8 | Dihedral group:D8 | Cyclic group:Z2 | Klein four-group |

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

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

Center of M16 | M16 | Cyclic group:Z4 | Klein four-group |

Center of direct product of D8 and Z2 | Direct product of D8 and Z2 | Klein four-group | Klein four-group |

## Relation with other properties

### Stronger properties

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

marginal subgroup of abelian group | |FULL LIST, MORE INFO |

### Weaker properties

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

abelian characteristic subgroup | abelian and a characteristic subgroup: invariant under all automorphisms | |FULL LIST, MORE INFO | ||

abelian normal subgroup | abelian and a normal subgroup: invariant under all inner automorphisms | |FULL LIST, MORE INFO |