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.