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

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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

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 H of a group G is termed a normal subgroup whose center is contained in the center of the whole group if ...
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. H is normal in G and Z(H) \le Z(G), where Z(H) and Z(G) are the respective centers of H and G.
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 g \in G, the automorphism x \mapsto gxg^{-1} restricts to a center-fixing automorphism of the subgroup H, i.e., it sends H to itself and fixes every element of Z(H).

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