Right-quotient-transitively central factor implies join-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., right-quotient-transitively central factor) must also satisfy the second subgroup property (i.e., join-transitively central factor)
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Statement with symbols
Proof: Let , and consider the quotient map . Then, .
- is a central factor of : By fact (1), since is a central factor of , is a central factor of , which translates to the above.
- is a central factor of : This follows from the assumption about in the given data.
This completes the proof.