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

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

• Join-transitively central factor not implies right-quotient-transitively central factor

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.