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