Group with unique Schur covering group

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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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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)