Baer Lie property is not quotient-closed

This article gives the statement, and possibly proof, of a group property (i.e., Baer Lie group) not satisfying a group metaproperty (i.e., quotient-closed group property).
View all group metaproperty dissatisfactions | View all group metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for group properties
Get more facts about Baer Lie group|Get more facts about quotient-closed group property|

Statement

It is possible to have a Baer Lie group $G$ and $H$ a normal subgroup of $G$ such that the quotient group $G/H$ is not a Baer Lie group.

Related facts

• Lazard Lie property is not quotient-closed

Proof

Further information: unitriangular matrix group:UT(3,Q), quotient of UT(3,Q) by a central Z

Suppose $G$ is the group $UT(3,\mathbb{Q})$, the unitriangular matrix group of degree three over the field of rational numbers. Let $H$ be a central subgroup of $G$ that is isomorphic to $\mathbb{Z}$, the group of integers. The quotient group $G/H=UT(3,\mathbb{Q})/\mathbb{Z}$ is not a Baer Lie group: it is a group of nilpotency class exactly two that has 2-torsion.