Direct factor implies join-transitively central factor

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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., direct factor) must also satisfy the second subgroup property (i.e., join-transitively central factor)
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Any direct factor of a group is a join-transitively central factor: its join with any central factor is a central factor.

Facts used

  1. Direct factor implies right-quotient-transitively central factor
  2. Right-quotient-transitively central factor implies join-transitively central factor


The proof follows by piecing together facts (1) and (2).