# Normal subgroup whose center is contained in the center of the whole group

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup whose center is contained in the center of the whole group

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed a **normal subgroup whose center is contained in the center of the whole group** if it satisfies the following equivalent conditions.

No. | Shorthand | A subgroup of a group is termed a normal subgroup whose center is contained in the center of the whole group if ... | A subgroup of a group is termed a normal subgroup whose center is contained in the center of the whole group if ... |
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1 | normal, and center contained in whole group | it is a normal subgroup and is also a subgroup whose center is contained in the center of the whole group. | is normal in and , where and are the respective centers of and . |

2 | inner automorphism restricts to center-fixing automorphism | any inner automorphism of the whole group restricts to a center-fixing automorphism of the subgroup. | for any , the automorphism restricts to a center-fixing automorphism of the subgroup , i.e., it sends to itself and fixes every element of . |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

center-fixing automorphism-balanced subgroup | every center-fixing automorphism of the whole group restricts to a center-fixing automorphism of the subgroup. | |FULL LIST, MORE INFO | ||

central factor | every inner automorphism restricts to an inner automorphism | |FULL LIST, MORE INFO | ||

conjugacy-closed normal subgroup | every inner automorphism restricts to a class-preserving automorphism | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup | every inner automorphism sends the subgroup to itself | |FULL LIST, MORE INFO | ||

subgroup whose center is contained in the center of the whole group | its center is contained in the whole group's center. | |FULL LIST, MORE INFO |

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.

Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression | is a normal subgroup whose center is contained in the center of the whole group in if ... | This means that being a normal subgroup whose center is contained in the center of the whole group is ... | Additional comments |
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inner automorphism center-fixing automorphism | every inner automorphism of restricts to a center-fixing automorphism of | a left-inner subgroup property. |