Central factor implies normal
From Groupprops
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., central factor) must also satisfy the second subgroup property (i.e., normal subgroup)
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Contents
Statement
Verbal statement
Any central factor of a group is a normal subgroup.
Property-theoretic statement
The subgroup property of being a central factor is stronger than the subgroup property of being a normal subgroup.
Definitions used
Normal subgroup
Further information: Normal subgroup
- Definition in terms of function restriction expressions
- Definition in terms of normalizer
Central factor
Further information: Central factor
- Definition in terms of function restriction expression
- Definition in terms of centralizer
Proof
Using function restriction expressions
This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic
The function restriction expression for normality is:
Inner automorphism Function
The function restriction expression for central factor is:
Inner automorphism Inner automorphism
Since the left sides in both cases are the same, and the right side for central factor is stronger than the right side for normality, the property of being a central factor is stronger than the property of being a normal subgroup.