# Abnormal normalizer and 2-subnormal not implies normal

From Groupprops

## Contents

## Statement

A 2-subnormal subgroup (?) with abnormal normalizer, i.e., a 2-subnormal subgroup whose normalizer in the whole group is an abnormal subgroup, need not be a normal subgroup.

## Related facts

### Corollaries

## Proof

### An example of the symmetric group on four letters

`Further information: symmetric group:S4`

Let by the symmetric group on the set , and be the two-element subgroup generated by the double transposition . Then, the normalizer is the dihedral group of order eight, which is a maximal non-normal subgroup, and hence abnormal. Hence, has abnormal normalizer.

On the other hand, is contained in the subgroup of order four, comprising the identity and the double transpositions. is normal in , and is normal in , so is 2-subnormal in .