Complemented normal subgroup

Equivalence of definitions
Note that any complement to $$H$$ in $$G$$ must be isomorphic to the quotient group $$G/H$$.

Extreme examples

 * Every group is a complemented normal subgroup of itself, with the complement being the trivial subgroup.
 * The trivial subgroup is a complemented normal subgroup in any group, with the complement being the whole group.

Examples in abelian groups
If the whole group is an abelian group, being a complemented normal subgroup is equivalent to being a direct factor, i.e., a part of an internal direct product.

General examples

 * Low occurrence: A splitting-simple group is a nontrivial group with no proper nontrivial complemented normal subgroup. Any simple group is splitting-simple, but there exist splitting-simple groups that are not simple.
 * High occurrence: A C-group is a group in which every subgroup is permutably complemented, and hence, every normal subgroup is a complemented normal subgroup.

Metaproperties
Note that the notation $$K$$ as used here is not to be confused with the $$K$$ used to denote the complement in the definition as presented above.

Related properties
Retract is a (not necessarily normal) subgroup that has a permutable complement which is a normal subgroup.

Metaproperties
The intersection of two complemented normal subgroups need not be a complemented normal subgroup. The proof of this relies on the same example which shows that direct factor is not intersection-closed.

If $$H$$ is a complemented normal subgroup in $$G$$, and $$M$$ is an intermediate subgroup, then $$H$$ is a complemented normal subgroup in $$M$$. In fact, the retraction for $$M$$ is simply the restriction to $$M$$ of the retraction on $$G$$. To prove that this retraction actually restricts to a well-defined map on $$M$$, we need to use the fact that $$M$$ contains $$H$$.

Suppose $$H \le K \le G$$ are groups such that $$H$$ is a complemented normal subgroup of $$G$$ and $$K/H$$ is a complemented normal subgroup of $$G/H$$. Then, $$K$$ is a complemented normal subgroup of $$G$$.