# Residually nilpotent group with abelianization that is divisible by a prime need not be divisible by that prime

From Groupprops

## Statement

It is possible to have a residually nilpotent group and a prime number such that the abelianization of is -divisible, but itself is not -divisible.

## Proof

`Further information: infinite dihedral group`

Let be the infinite dihedral group:

and let be any prime number other than 2. Then:

- is residually nilpotent: The member of lower central series of for , is . The intersection of these is trivial.
- The abelianization of is -divisible: The abelianization of is a Klein four-group, which is divisible for all odd primes.
- is not -divisible: The element has no roots.