# Complemented 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., complemented normal subgroup) neednotsatisfy the second subgroup property (i.e., direct factor)

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## Statement

It is possible to have a group and a complemented normal subgroup such that is not a direct factor of . In other words, has a permutable complement in , but no normal complement in .

## Examples

Here are some examples where the ambient group is quite important and easy to understand:

Group part | Subgroup part | Quotient part | |
---|---|---|---|

A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |

Here are some examples where the ambient group is somewhat more complicated:

Here are some examples where the ambient group is even more complicated:

## Proof

`Further information: dihedral group:D8, cyclic maximal subgroup of dihedral group:D8`

We can take to be the dihedral group given as:

and to be its unique cyclic maximal subgroup . Then is a permutable complement to in , but has no normal complement, so is not a direct factor.