Central factor implies normal

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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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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

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 \to Function

The function restriction expression for central factor is:

Inner automorphism \to 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.

Proof using definition in terms of centralizer

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