# Fully invariant direct factor implies left-transitively homomorph-containing

From Groupprops

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., fully invariant direct factor) must also satisfy the second subgroup property (i.e., left-transitively homomorph-containing subgroup)

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## Statement

Suppose is a fully invariant direct factor of a group . Then, is a left-transitively homomorph-containing subgroup of : for any group in which is a homomorph-containing subgroup, is also a homomorph-containing subgroup.