# Solvable 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 solvable group and a prime number such that the abelianization of is -divisible, but itself is not -divisible.

## Related facts

- Residually nilpotent group with abelianization that is divisible by a prime need not be divisible by that prime
- Equivalence of definitions of nilpotent group that is divisible for a set of primes

## Proof

`Further information: infinite dihedral group`

Let be a infinite dihedral group:

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

- is solvable: In fact, is metacyclic. It has a cyclic normal subgroup with a cyclic quotient group isomorphic to cyclic group:Z2 (generated by the image of ).
- 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.