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

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## Contents

## Statement

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

## Related facts

- For finite groups, being a group for which every subgroup is Schur-trivial is equivalent to being a finite group with periodic cohomology, which in turn is equivalent to every abelian subgroup being cyclic.

## 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 .
- is abstractly isomorphic to dihedral group:D8, which is
*not*Schur-trivial.

### Example of general linear group

We could take:

- is general linear group:GL(2,3), which is a Schur-trivial group of order 48.
- is the subgroup D8 in GL(2,3), obtained by using the faithful irreducible representation of dihedral group:D8 over field:F3.
- is isomorphic to dihedral group:D8, which is not Schur-trivial.