Schur multiplier of divisible nilpotent group need not be divisible by any prime
It is possible to have a divisible nilpotent group (i.e., is a nilpotent group and it is divisible by all primes) such that the Schur multiplier is not a divisible abelian group. In fact, we can choose an example where is a group of nilpotency class two and the Schur multiplier is not divisible by any prime.
Further information: quotient of UT(3,Q) by a central Z
The group described here is a quotient group of unitriangular matrix group:UT(3,Q) by a central subgroup isomorphic to the group of integers, which we can think of as a Z in Q inside the center, which is a copy of . Explicitly, it is matrices of the form:
with the matrix multiplication defined as:
where is understood to be the image of under the quotient map .
- MathOverflow question: Homology groups of divisible and powered (nilpotent) groups: The original question (part (1)) describes the group, and the answer by Ralph explains the homology computation.