# Join of direct factor and 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

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

- There exist subgroups of such that is a direct factor of , is a central subgroup of , and is the join (in this case, also the product) of and .
- There exists a direct factor of contained in such that is a central subgroup of the quotient group .

## Relation with other properties

### Stronger properties

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

Direct factor | obvious | |FULL LIST, MORE INFO | ||

Central subgroup | obvious | |FULL LIST, MORE INFO |

### Weaker properties

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

Central factor | product with its centralizer is the whole group | |FULL LIST, MORE INFO | ||

Join-transitively central factor | join with any central factor is a central factor | |FULL LIST, MORE INFO | ||

Direct factor over central subgroup | image in quotient by center is a direct factor | |FULL LIST, MORE INFO |