Direct factor implies right-quotient-transitively central factor
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., direct factor) must also satisfy the second subgroup property (i.e., right-quotient-transitively central factor)
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Statement with symbols
Given: A group , a direct factor of , contains and is a central factor of .
To prove: is a central factor of .
Proof: Since is a direct factor of , there exists a normal complement to in , with and trivial. Let .
Consider the map that sends every element of to its -coset in . This map is an isomorphism, since the kernel is trivial, and . Thus, .
- Every element of centralizes every element of : Since both and are normal in , is contained in both, and since they intersect trivially, is trivial, so every element of centralizes every element of .
- : Since is an isomorphism, and is a central factor of , is a central factor of .
- centralizes , i.e., : By step (1), centralizes , since centralizes . It also centralizes by definition. Thus, it centralizes .
- : We have .
- : This follows from the previous two steps.
This completes the proof.