# Schur-trivial group

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

## Definition

A group is said to be Schur-trivial or a group with trivial Schur multiplier if it satisfies the following equivalent conditions:

1. Its Schur multiplier is the trivial group.
2. The homomorphism from its exterior square to its derived subgroup defined by the commutator map is an isomorphism of groups to the derived subgroup.
3. It is isomorphic to its own Schur covering group with the covering map being the identity map.

## Examples

### Groups satisfying the property

Here are some basic/important groups satisfying the property:

GAP ID
Cyclic group:Z22 (1)
Symmetric group:S36 (1)

Here are some relatively less basic/important groups satisfying the property:

GAP ID
Quaternion group8 (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:Q1616 (9)
M1616 (6)
Semidihedral group:SD1616 (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:A560 (5)
Dihedral group:D88 (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 $G$ and a subgroup $H$ of $G$ such that $H$ is not Schur-trivial. (For a finite group, every subgroup being Schur-trivial is equivalent to the group being a finite group with periodic cohomology).
characteristic subgroup-closed group property No Schur-triviality is not characteristic subgroup-closed It is possible to have a Schur-trivial group $G$ and a characteristic subgroup $H$ of $G$ such that $H$ is not Schur-trivial.
quotient-closed group property No Schur-triviality is not quotient-closed It is possible to have a Schur-trivial group $G$ and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ is not Schur-trivial.
finite direct product-closed group property No Schur-triviality is not finite direct product-closed It is possible to have two Schur-trivial groups $G_1$ and $G_2$ such that the external direct product $G_1 \times G_2$ is not Schur-trivial.
isoclinism-invariant group property No Schur-triviality is not isoclinism-invariant It is possible to have isoclinic groups $G_1$ and $G_2$ such that $G_1$ is Schur-trivial but $G_2$ is not 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
group with unique Schur covering group The $\operatorname{Ext}^1$ of the abelianization over the Schur multiplier is trivial. |FULL LIST, MORE INFO