# Center-fixing automorphism-balanced subgroup

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Definition

The following are equivalent definitions of center-fixing automorphism-balanced subgroup.

No. | Shorthand | A subgroup of a group is termed a center-fixing automorphism-balanced subgroup if ... | A subgroup of a group is termed a center-fixing automorphism-balanced subgroup if ... |
---|---|---|---|

1 | center-fixing automorphism-invariant, and center contained in group's center | it is a center-fixing automorphism-invariant subgroup of the whole group, and its center is contained in the center of the whole group. | is invariant under any automorphism of with the property that for all , and additionally, . Here and denote respectively the centers of and . |

2 | center-fixing automorphism restricts to center-fixing automorphism | any center-fixing automorphism of the whole group restricts to a center-fixing automorphism of the subgroup. | For any automorphism of with the property that for all , we have that , and also that for all . |

## Relation with other properties

### Weaker properties

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

center-fixing automorphism-invariant subgroup | invariant under all center-fixing automorphisms | |||

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

normal subgroup | every inner automorphism sends the subgroup to itself | Center-fixing automorphism-invariant subgroup, Normal subgroup whose center is contained in the center of the whole group|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. | Normal subgroup whose center is contained in the center of the whole group|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 center-fixing automorphism-balanced subgroup of if ... | This means that being center-fixing automorphism-balanced is ... | Additional comments |
---|---|---|---|

center-fixing automorphism center-fixing automorphism | every center-fixing automorphism of restricts to a center-fixing automorphism of | the balanced subgroup property for center-fixing automorphisms | Hence, it is a t.i. subgroup property, both transitive and identity-true |