Homomorph-containment is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., homomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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Get more facts about homomorph-containing subgroup |Get facts that use property satisfaction of homomorph-containing subgroup | Get facts that use property satisfaction of homomorph-containing subgroup|Get more facts about quotient-transitive subgroup property

Statement

Statement with symbols

Suppose $G$ is a group, and $H\leq K\leq G$ are subgroups of $G$ such that $H$ is a homomorph-containing subgroup of $G$ and $K/H$ is a homomorph-containing subgroup of $G/H$. Then, $K$ is a homomorph-containing subgroup of $G$.

Related facts

Related facts about homomorph-containing subgroups

• Homomorph-containment is not transitive
• Homomorph-containment satisfies intermediate subgroup condition

Related facts about quotient-transitivity

• Full invariance is quotient-transitive
• Strict characteristicity is quotient-transitive
• Characteristicity is quotient-transitive
• Normality is quotient-transitive