Full invariance is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about fully invariant subgroup |Get facts that use property satisfaction of fully invariant subgroup | Get facts that use property satisfaction of fully invariant subgroup|Get more facts about quotient-transitive subgroup property

Statement

Statement with symbols

Suppose $H\leq K\leq G$ are groups, such that $H$ is a fully invariant subgroup of $G$ and $K/H$ is a fully invariant subgroup of $G/H$. Then, $K$ is a fully invariant subgroup of $G$.

Related facts

Generalization and other particular cases

A generalization of this fact is:

• Quotient-balanced implies quotient-transitive

Other instances of this generalization are:

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

Other similar facts

• Homomorph-containment is quotient-transitive
• Subhomomorph-containment is quotient-transitive