# Join of direct factor and central subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Definition with symbols

A subgroup $H$ of a group $G$ is termed a join of direct factor and central subgroup or a central subgroup over direct factor if it satisfies the following equivalent conditions:

1. There exist subgroups $A,B$ of $G$ such that $A$ is a direct factor of $G$, $B$ is a central subgroup of $G$, and $H$ is the join (in this case, also the product) of $A$ and $B$.
2. There exists a direct factor $A$ of $G$ contained in $H$ such that $H/A$ is a central subgroup of the quotient group $G/A$.

## Relation with other properties

### Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions