Abelian direct factor
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This article describes a property that arises as the conjunction of a subgroup property: direct factor with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: central subgroup and direct factor
View other subgroup property conjunctions | view all subgroup properties
Definition
A subgroup of a group is an abelian direct factor or central direct factor or complemented central subgroup if it satisfies the following conditions:
- It is abelian as a group and is a direct factor of the whole group.
- It is both a central subgroup and a direct factor of the whole group.
- It is both a central subgroup and a permutably complemented subgroup of the whole group.
- It is both a central subgroup and a complemented normal subgroup of the whole group.
- It is both a central subgroup and a complemented central factor of the whole group.
- It is both a central subgroup and a lattice-complemented subgroup of the whole group.
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Central subgroup | contained in the center | |FULL LIST, MORE INFO | ||
| Direct factor | factor in an internal direct product | |FULL LIST, MORE INFO | ||
| Complemented central factor | central factor having a permutable complement | |FULL LIST, MORE INFO | ||
| Complemented normal subgroup | normal subgroup having a permutable complement | |FULL LIST, MORE INFO | ||
| Abelian hereditarily normal subgroup | |FULL LIST, MORE INFO | |||
| Abelian normal subgroup | |FULL LIST, MORE INFO | |||
| Permutably complemented subgroup | |FULL LIST, MORE INFO | |||
| Lattice-complemented subgroup | |FULL LIST, MORE INFO |