# Normal closure of 3-subnormal subgroup of prime order in nilpotent group need not be abelian

From Groupprops

## Statement

It is possible to have a nilpotent group and a 3-subnormal subgroup (?) of of order a prime number such that the Normal closure (?) of in is not an abelian group.

In fact, we can take itself to be a finite p-group.

## Related facts

- Normal closure of 2-subnormal subgroup of prime order in nilpotent group is abelian
- Normal closure of 2-subnormal subgroup of prime power order in nilpotent group has nilpotency class at most equal to prime-base logarithm of order

## Proof

### Example of the dihedral group of order 16

`Further information: dihedral group:D16, subgroup structure of dihedral group:D16`

Let be the dihedral group of order 16 (degree 8), given explicitly by the presentation:

Suppose is the subgroup of . Then:

- is a subgroup of order 2 in .
- is 3-subnormal in .
- The normal closure of in is D8 in D16, the subgroup , which is isomorphic to dihedral group:D8. This is not an abelian group.