Abelian direct factor

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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:

  1. It is abelian as a group and is a direct factor of the whole group.
  2. It is both a central subgroup and a direct factor of the whole group.
  3. It is both a central subgroup and a permutably complemented subgroup of the whole group.
  4. It is both a central subgroup and a complemented normal subgroup of the whole group.
  5. It is both a central subgroup and a complemented central factor of the whole group.
  6. 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 Direct factor|FULL LIST, MORE INFO
Complemented normal subgroup normal subgroup having a permutable complement Direct factor|FULL LIST, MORE INFO
Abelian hereditarily normal subgroup |FULL LIST, MORE INFO
Abelian normal subgroup Central subgroup|FULL LIST, MORE INFO
Permutably complemented subgroup Direct factor|FULL LIST, MORE INFO
Lattice-complemented subgroup Direct factor|FULL LIST, MORE INFO