# Subgroup whose center is contained in the center of the whole group

From Groupprops

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]

## Contents

## Definition

The notion of **subgroup whose center is contained in the center of the whole group** has the following equivalent definitions:

No. | Shorthand | A subgroup of a group is termed a subgroup whose center is contained in the center of the whole group if ... | A subgroup of a group is termed a subgroup whose center is contained in the center of the whole group if ... |
---|---|---|---|

1 | center containment | the center of the subgroup is contained in the center of the whole group. | where denotes the center of and denotes the center of . |

2 | center equals intersection with center | the center of the subgroup equals the intersection of the subgroup with the center of the whole group. | where denotes the center of and denotes the center of . |

3 | intersection with center equals intersection with centralizer | the intersection of the subgroup with the center of the whole group equals the intersection of the subgroup with its centralizer. | where is the centralizer of in and is the center of . |

4 | inclusion induces well-defined map on inner automorphism groups | the inclusion of the subgroup in the whole group induces a homomorphism of the inner automorphism groups (in fact, this homomorphism must be injective). | the subgroup inclusion induces a well-defined map , or in other words, a well-defined map . |

This definition is presented using a tabular format. |View all pages with definitions in tabular format

## Examples

### Extreme examples

- The trivial subgroup in any group satisfies this property.
- Every group satisfies this property in itself.

### High occurrence examples

- If the whole group is an abelian group, every subgroup satisfies the property.
- A centerless group satisfies this property as a subgroup in any bigger group.

### Subgroups satisfying the property

Here are some examples of subgroups in basic/important groups satisfying the property:

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

### Subgroups dissatisfying the property

Here are some examples of subgroups in basic/important groups *not* satisfying the property:

Here are some some examples of subgroups in relatively less basic/important groups *not* satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups *not* satisfying the property:

## Metaproperties

Metaproperty | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

transitive subgroup property | Yes | (link pending) | If are groups such that satisfies this property in and satisfies this property in , then satisfies this property in . |

trim subgroup property | Yes | In any group , the whole group and the trivial subgroup satisfy this property. |

## Relation with other properties

### Stronger properties

### Dual property

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