Schur-triviality is not subgroup-closed: Difference between revisions
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We take the following: | We take the following: | ||
* <math>G</math> is [[semidihedral group:SD16]], which is a Schur-trivial group. | * <math>G</math> is [[particular example::semidihedral group:SD16]], which is a Schur-trivial group. | ||
* <math>H</math> is the subgroup [[D8 in SD16]] inside <math>G</math>. | * <math>H</math> is the subgroup [[particular example::D8 in SD16]] inside <math>G</math>. | ||
* <math>H</math> is abstractly isomorphic to [[dihedral group:D8]], which is ''not'' Schur-trivial. | * <math>H</math> is abstractly isomorphic to [[particular example::dihedral group:D8]], which is ''not'' Schur-trivial. | ||
===Example of general linear group=== | ===Example of general linear group=== | ||
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We could take: | We could take: | ||
* <math>G</math> is [[general linear group:GL(2,3)]], which is a Schur-trivial group of order 48. | * <math>G</math> is [[particular example::general linear group:GL(2,3)]], which is a Schur-trivial group of order 48. | ||
* <math>H</math> is the subgroup [[D8 in GL(2,3)]], obtained by using the [[faithful irreducible representation of dihedral group:D8]] over [[field:F3]]. | * <math>H</math> is the subgroup [[particular example::D8 in GL(2,3)]], obtained by using the [[faithful irreducible representation of dihedral group:D8]] over [[field:F3]]. | ||
* <math>H</math> is isomorphic to [[dihedral group:D8]], which is not Schur- | * <math>H</math> is isomorphic to [[particular example::dihedral group:D8]], which is not Schur-trivial. | ||
Latest revision as of 23:42, 12 January 2013
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).
View all group metaproperty dissatisfactions | View all group metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for group properties
Get more facts about Schur-trivial group|Get more facts about subgroup-closed group property|
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.