# Schur multiplier of divisible nilpotent group need not be divisible by any prime

## Statement

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.

## Related facts

### Opposite facts

## Proof

`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 .

The Schur multiplier of turns out to be isomorphic to the group of integers , which is not divisible by any prime.

## References

- 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.