Group with unique Schur covering group

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Statement

A group with unique Schur covering group is a group $G$ satisfying the following equivalent conditions:

1. It has a unique Schur covering group, i.e., given Schur covering groups $K_{1},K_{2}$ of $G$ with covering maps $\pi _{1}:K_{1}\to G$ and $\pi _{2}:K_{2}\to G$, there is an isomorphism $\theta :K_{1}\to K_{2}$ such that $\pi _{1}=\pi _{2}\circ \theta$.
2. $\operatorname {Ext} ^{1}(G^{\operatorname {ab} },M(G))=0$ where $G^{\operatorname {ab} }$ is the abelianization of $G$ (and also identified with $H_{1}(G;\mathbb {Z} )$) while $M(G)$ is the Schur multiplier of $G$ (and also identified with $H_{2}(G;\mathbb {Z} )$). Note that for $G$ a finite group, this is equivalent to verifying that $G^{\operatorname {ab} }$ and $M(G)$ have relatively prime orders.

Examples

The smallest example of a group $G$ where neither the abelianization nor the Schur multiplier is trivial, but the Schur covering group is still unique, is alternating group:A4, whose unique Schur multiplier is special linear group:SL(2,3).

In addition, we can take perfect groups and Schur-trivial groups as examples.

Non-examples

The smallest non-example is the Klein four-group, which has two Schur covering groups (dihedral group:D8 and quaternion group). Note that in general the number of Schur covering groups is $|\operatorname {Ext} ^{1}(G^{\operatorname {ab} },M(G))|$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
perfect group	the abelianization is trivial.		(any Schur-trivial group that is not perfect -- such as the quaternion group) or (alternating group:A4) gives a counterexample	|FULL LIST, MORE INFO
Schur-trivial group	the Schur multiplier is trivial		(any perfect group that is not Schur-trivial -- such as alternating group:A5) or (alternating group:A4) gives a counterexample	|FULL LIST, MORE INFO
cyclic group		(via Schur-trivial)	(via Schur-trivial)
simple non-abelian group		(via perfect)	(via perfect)