# Normal not implies direct factor

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., direct factor)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not direct factor|View examples of subgroups satisfying property normal subgroup and direct factor

## Statement

A normal subgroup of a group need not be a direct factor.

## Proof

Let $G$ be cyclic group of order four and $H$ be the unique subgroup of order two, comprising the squares (or the elements whose order divides two). Then:

• $H$ is a normal subgroup of $G$.
• $H$ is not a direct factor of $G$: In fact, it is the only proper nontrivial subgroup of $G$.