Right-quotient-transitively central 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., right-quotient-transitively central factor) must also satisfy the second subgroup property (i.e., join-transitively central factor)
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Statement

Verbal statement

Any right-quotient-transitively central factor is a join-transitively central factor.

Statement with symbols

Suppose H is a normal subgroup of a group G such that for any subgroup K of G containing H, such that K/H is a central factor of G/H, K is also a central factor of G.

Then, for any central factor N of G, the join of subgroups \langle H,N \rangle, which in this case is also the product of subgroups HN is also a central factor.

Related facts

Converse

Facts used

  1. Central factor satisfies image condition

Proof

Given: H is a normal subgroup of a group G such that for any subgroup K of G containing H, such that K/H is a central factor of G/H, K is also a central factor of G.

To prove: For any central factor N of G, the join of subgroups \langle H,N \rangle, which in this case is also the product of subgroups HN is also a central factor.

Proof: Let K = HN, and consider the quotient map \rho:G \to G/H. Then, \rho(N) = K/H.

  1. K/H is a central factor of G/H: By fact (1), since N is a central factor of G, \rho(N) is a central factor of \rho(G), which translates to the above.
  2. K is a central factor of G: This follows from the assumption about H in the given data.

This completes the proof.