# Schur-triviality is not quotient-closed

This article gives the statement, and possibly proof, of a group property (i.e., Schur-trivial group) not satisfying a group metaproperty (i.e., quotient-closed group property).
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## Statement

It is possible to have a Schur-trivial group $G$ (i.e., the Schur multiplier of $G$ is the trivial group) and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ is not Schur-trivial.

## Proof

### Example of the quaternion group

This is the smallest example:

• $G$ is the quaternion group, which is a Schur-trivial group.
• $H$ is the center of quaternion group, which is of order two, and is isomorphic to cyclic group:Z2.
• $G/H$ is isomorphic to the Klein four-group, which is not Schur-trivial.

### Perfect group examples

We can construct many examples this way: let $G$ be the universal central extension of some perfect group $M$ that is not Schur-trivial. Then, $G$ is a Schur-trivial group, and $M$ is the quotient of $G$ by some central subgroup, and is not Schur-trivial.