# Schur-triviality is not characteristic subgroup-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., characteristic subgroup-closed group property).

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## Statement

It is possible to have a Schur-trivial group and a characteristic subgroup of such that is not a Schur-trivial group.

## Proof

### Example of semidihedral group

We take the following:

- is semidihedral group:SD16, which is a Schur-trivial group.
- is the subgroup D8 in SD16 inside . This is an isomorph-free subgroup, hence is characteristic.
- is abstractly isomorphic to dihedral group:D8, which is
*not*Schur-trivial.