Homomorph-containment is strongly join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., homomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., strongly join-closed subgroup property)
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Statement

Statement with symbols

Suppose $G$ is a group, $I$ is an indexing set, and $H_{i},i\in I$ , is a collection of homomorph-containing subgroups of $G$ . Then, the join of subgroups $\langle H_{i}\rangle _{i\in I}$ is also a homomorph-containing subgroup of $G$ .

Related facts

Related facts about join-closed

Proof

Given: A group $G$ , an indexing set $I$ , a collection $H_{i}$ of homomorph-containing subgroups of $G$ , $i\in I$ . $H=\langle H_{i}\rangle _{i\in I}$ . A homomorphism $\varphi :H\to G$ .

To prove: $\varphi (H)$ is containined in $H$ .

Proof: $\varphi (H)=\varphi \langle H_{i}\rangle _{i\in I}=\langle \varphi (H_{i})\rangle _{i\in I}$ . Since each $H_{i}$ is homomorph-containing in $G$ , $\varphi (H_{i})$ is contained in $H_{i}$ , so the join of $\varphi (H_{i}),i\in I$ , is contained in the join of the $H_{i}$ s, which is $H$ .