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