Group with unique Schur covering group
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A group with unique Schur covering group is a group satisfying the following equivalent conditions:
- It has a unique Schur covering group, i.e., given Schur covering groups of with covering maps and , there is an isomorphism such that .
- where is the abelianization of (and also identified with ) while is the Schur multiplier of (and also identified with ). Note that for a finite group, this is equivalent to verifying that and have relatively prime orders.
The smallest example of a group 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).
Relation with other 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)|