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 ![]() ![]() |
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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. | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 ![]() ![]() ![]() ![]() ![]() |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
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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 | ![]() ![]() |
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 ![]() |
every inner automorphism of ![]() ![]() |
a left-inner subgroup property. |