Nilpotent and torsion-free not implies torsion-free abelianization
This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., derived subgroup) does not always satisfy a particular subgroup property (i.e., quotient-torsion-freeness-closed subgroup)
View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions
Somewhat surprisingly, the dual fact to this is not true. The dual fact, if true, would state that the center of a divisible nilpotent group need not be divisible (and in particular, that the center need not be divisibility-closed in a nilpotent group). This is false. In fact, upper central series members are completely divisibility-closed in nilpotent group.
Further information: central product of UT(3,Z) and Q
- is torsion-free.
- is isomorphic to , and is isomorphic to . This has -torsion for all primes .