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

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 $G$ 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.

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 $\operatorname{Ext}^1$ of the abelianization over the Schur multiplier is trivial.			|FULL LIST, MORE INFO