# Nilpotent group need not be embeddable in a divisible nilpotent group

## Statement

It is possible to have a nilpotent group $G$ such that $G$ cannot be embedded in any divisible nilpotent group.

## Related facts

The result can be understood in a 2 X 2 matrix of results:

Group assumption on both the starting group and the big group Divisibility/powering/torsion assumption on the starting group Divisibility/powering/torsion assumption on the big group Is this always possible? Proof
nilpotent group none divisible group No nilpotent group need not be embeddable in a divisible nilpotent group
nilpotent group $\pi$-torsion-free group (equivalently, $\pi$-powering-injective group) $\pi$-powered group Yes every pi-torsion-free nilpotent group can be embedded in a unique minimal pi-powered nilpotent group
arbitrary group none divisible group Yes every group is a subgroup of a divisible group, every pi-group is a subgroup of a divisible pi-group
arbitrary group $\pi$-torsion-free group (equivalently, $\pi$-powering-injective group) $\pi$-powered group No Powering-injective group need not be embeddable in a rationally powered group

## References

We remark that an arbitrary nilpotent group cannot always be embedded in a divisible group which is nilpotent. For a theorem of Chernikov [1] asserts that the elements of finite order in a divisible group with an ascending central series form a subgroup of the centre and therefore, in particular, the quaternion group of order 8 cannot be embedded in a nilpotent divisible group.