# Schur-triviality is not quotient-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., Schur-trivial group)notsatisfying a group metaproperty (i.e., quotient-closed group property).

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## Statement

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

## Proof

### Example of the quaternion group

This is the smallest example:

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

### Perfect group examples

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