Central factor implies transitively normal
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., transitively normal subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about central factor|Get more facts about transitively normal subgroup
Statement with symbols
Given: A group , a central factor of , a normal subgroup of . is an element of and is an element of .
To prove: .
Proof: Let be the inner automorphism obtained as conjugation by .
- There exists such that the inner automorphism by , as an inner automorphism of , is the restriction to of . In other words, for any , : This follows from the definition of central factor.
- : This follows from the fact that is normal in .
- : This follows immediately from the previous two steps.