# Direct factor over 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

Suppose $H$ is a subgroup of a group $G$. We say that $H$ is a direct factor over central subgroup of $G$ if it satisfies the following equivalent conditions:

1. There exists a subgroup $A$ of $H$ such that $A$ is a central subgroup of $G$ and $H/A$ is a direct factor of the quotient group $G/A$.
2. In the quotient map $\rho:G \to G/Z(G)$, where $Z(G)$ is the center of $G$, $\rho(H)$ is a direct factor of $\rho(G) = G/Z(G)$.

## Relation with other properties

### Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Direct factor Join of direct factor and central subgroup|FULL LIST, MORE INFO
Central subgroup Join of direct factor and central subgroup|FULL LIST, MORE INFO