# No common composition factor with quotient group not implies complemented

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 having no common composition factor with its quotient group) neednotsatisfy the second subgroup property (i.e., complemented normal subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

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## Statement

It is possible to have a finite group having a normal subgroup such that and have no common composition factors, but is *not* a permutably complemented subgroup of .

## Related facts

### Similar facts

### Opposite facts

## Proof

`Further information: special linear group:SL(2,5)`

Let , the special linear group of matrices over the field of five elements. Let . Then, is a subgroup of order two and is isomorphic to the alternating group of degree five, which is simple. Thus, and have no common composition factors. However, has no complement in .