Join of finitely many direct factors implies 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., join of finitely many direct factors) must also satisfy the second subgroup property (i.e., central factor)
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Statement

Statement with symbols

Suppose $G$ is a group and $H_{1},H_{2},\dots ,H_{n}$ $n\geq 0$) are direct factors of $G$. Then, the join of the $H_{i}$s, which is also the product $H_{1}H_{2}\dots H_{n}$ (because direct factor implies normal) is a central factor of $G$.

Facts used

1. Trivial subgroup is central factor
2. Direct factor implies join-transitively central factor: The join of a direct factor and a central factor is a central factor.

Proof

Given: $G$ is a group and $H_{1},H_{2},\dots ,H_{n}$ are direct factors of $G$.

To prove: The join of all the $H_{i}$s is a central factor of $G$.

Proof: We prove this by induction on $n$, starting with the case $n=0$. In the case $n=0$, the join is the trivial subgroup, so fact (1) completes the proof.

Suppose the statement is true for $n-1$, and we want to prove it for $n$. Since the statement is true for $n-1$, we know that the join of $H_{1},H_{2},\dots ,H_{n-1}$ is a central factor. By fact (2), the join of this with $H_{n}$ must also be a central factor. But this latter join is the same as the join of $H_{1},H_{2},\dots ,H_{n}$. This completes the proof.