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 with symbols
Suppose is a group, is an indexing set, and , is a collection of homomorph-containing subgroups of . Then, the join of subgroups is also a homomorph-containing subgroup of .
Related facts about join-closed
- Isomorph-containment is strongly join-closed
- Isomorph-freeness is strongly join-closed
- Full invariance is strongly join-closed
- Characteristicity is strongly join-closed
- Normality is strongly join-closed
Given: A group , an indexing set , a collection of homomorph-containing subgroups of , . . A homomorphism .
To prove: is containined in .
Proof: . Since each is homomorph-containing in , is contained in , so the join of , is contained in the join of the s, which is .