Schur-trivial group
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Contents
Definition
A group is said to be Schur-trivial or a group with trivial Schur multiplier if it satisfies the following equivalent conditions:
- Its Schur multiplier is the trivial group.
- The homomorphism from its exterior square to its derived subgroup defined by the commutator map is an isomorphism of groups to the derived subgroup.
- It is isomorphic to its own Schur covering group with the covering map being the identity map.
Examples
Extreme examples
- The trivial group is Schur-trivial.
- Cyclic groups are Schur-trivial.
Groups satisfying the property
Here are some basic/important groups satisfying the property:
GAP ID | |
---|---|
Cyclic group:Z2 | 2 (1) |
Symmetric group:S3 | 6 (1) |
Here are some relatively less basic/important groups satisfying the property:
GAP ID | |
---|---|
Quaternion group | 8 (4) |
Special linear group:SL(2,3) | 24 (3) |
Special linear group:SL(2,5) | 120 (5) |
Here are some even more complicated/less basic groups satisfying the property:
GAP ID | |
---|---|
Generalized quaternion group:Q16 | 16 (9) |
M16 | 16 (6) |
Semidihedral group:SD16 | 16 (8) |
Groups dissatisfying the property
Here are some basic/important groups that do not satisfy the property:
Here are some relatively less basic/important groups that do not satisfy the property:
GAP ID | |
---|---|
Alternating group:A5 | 60 (5) |
Dihedral group:D8 | 8 (3) |
Here are some even more complicated/less basic groups that do not satisfy the property:
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | No | Schur-triviality is not subgroup-closed | It is possible to have a Schur-trivial group ![]() ![]() ![]() ![]() |
characteristic subgroup-closed group property | No | Schur-triviality is not characteristic subgroup-closed | It is possible to have a Schur-trivial group ![]() ![]() ![]() ![]() |
quotient-closed group property | No | Schur-triviality is not quotient-closed | It is possible to have a Schur-trivial group ![]() ![]() ![]() ![]() |
finite direct product-closed group property | No | Schur-triviality is not finite direct product-closed | It is possible to have two Schur-trivial groups ![]() ![]() ![]() |
isoclinism-invariant group property | No | Schur-triviality is not isoclinism-invariant | It is possible to have isoclinic groups ![]() ![]() ![]() ![]() |
Facts
- Finite group generated by Schur-trivial subgroups of relatively prime indices is Schur-trivial
- All Sylow subgroups are Schur-trivial implies Schur-trivial
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
cyclic group | generated by one element | Schur multiplier of cyclic group is trivial | the smallest non-cyclic group with trivial Schur multiplier is symmetric group:S3. | |FULL LIST, MORE INFO |
free group | has a freely generating set | Schur multiplier of free group is trivial | any finite cyclic group gives a counterexample. | |FULL LIST, MORE INFO |
Z-group | finite group in which every Sylow subgroup is cyclic | Z-group implies Schur-trivial | (infinite free groups; also, lots of counterexamples among finite groups, such as the quaternion group and various Schur covering groups of centerless groups. For instance, SL(2,3), SL(2,5)) | Finite group with periodic cohomology|FULL LIST, MORE INFO |
finite group with periodic cohomology | finite group in which every abelian subgroup is cyclic. Equivalently, every Sylow subgroup for odd primes is cyclic, and the 2-Sylow subgroup is cyclic or a generalized quaternion group | finite group with periodic cohomology is Schur-trivial | |FULL LIST, MORE INFO | |
superperfect group | perfect and Schur-trivial | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
group with unique Schur covering group | The ![]() |
|FULL LIST, MORE INFO |