Left-transitively complemented normal implies characteristic

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., left-transitively complemented normal subgroup) must also satisfy the second subgroup property (i.e., characteristic subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about left-transitively complemented normal subgroup|Get more facts about characteristic subgroup

Statement

Statement with symbols

Suppose $H$ is a subgroup of a group $K$ such that whenever $K$ is a complemented normal subgroup of some group $G$, $H$ is also a complemented normal subgroup of $G$. Then, $H$ is a characteristic subgroup of $K$. In particular, $H$ is a Complemented characteristic subgroup (?) of $K$.

Related facts

• Left transiter of normal is characteristic
• Left residual of normal by complemented normal equals characteristic

Converse

Obviously, the naive converse is not true, since a characteristic subgroup need not be a complemented normal subgroup. However, even if we consider complemented characteristic subgroups, they need not be left-transitively complemented normal. Further information: Complemented characteristic not implies left-transitively complemented normal

Facts used

1. Left residual of normal by complemented normal equals characteristic

Proof

Hands-on proof

Holomorph construction -- PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Proof using given facts

The proof follows directly from fact (1).