# Direct factor over central subgroup

From Groupprops

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

- There exists a subgroup of such that is a central subgroup of and is a direct factor of the quotient group .
- In the quotient map , where is the center of , is a direct factor of .

## 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 | |||

Join of direct factor and central subgroup | |FULL LIST, MORE INFO |

### Weaker properties

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

Normal subgroup |