# Normal not implies direct factor

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., direct factor)

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## Statement

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

## Related facts

- Characteristic not implies direct factor
- Normal not implies characteristic
- Central factor not implies direct factor

## Proof

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

- is a normal subgroup of .
- is
*not*a direct factor of : In fact, it is the only proper nontrivial subgroup of .