Schur multiplier
of a group G. It was introduced by Issai Schur (1904) in his work on projective representations.
of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, then the order of
Schur's original motivation for studying the multiplier was to classify projective representations of a group, and the modern formulation of his definition is the second cohomology group
, one takes a homomorphism into the projective general linear group
Schur (1904, 1907) showed that every finite group G has associated to it at least one finite group C, called a Schur cover, with the property that every projective representation of G can be lifted to an ordinary representation of C. The Schur cover is also known as a covering group or Darstellungsgruppe.
The Schur cover of a perfect group is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up to isoclinism.
[1] If the group G is finite and one considers only stem extensions, then there is a largest size for such a group C, and for every C of that size the subgroup K is isomorphic to the Schur multiplier of G. If the finite group G is moreover perfect, then C is unique up to isomorphism and is itself perfect.
Such C are often called universal perfect central extensions of G, or covering group (as it is a discrete analog of the universal covering space in topology).
, then the covering group itself can be presented in terms of F but with a smaller normal subgroup S, that is,
Because of this simplicity, expositions such as (Aschbacher 2000, §33) handle the perfect case first.
These are all slightly later results of Schur, who also gave a number of useful criteria for calculating them more explicitly.
The borderline case is thus quite interesting: finite groups with the same number of generators as relations are said to have a deficiency zero.
For a group to have deficiency zero, the group must have a trivial Schur multiplier because the minimum number of generators of the Schur multiplier is always less than or equal to the difference between the number of relations and the number of generators, which is the negative deficiency.
An efficient group is one where the Schur multiplier requires this number of generators.
[2] A fairly recent topic of research is to find efficient presentations for all finite simple groups with trivial Schur multipliers.
Such presentations are in some sense nice because they are usually short, but they are difficult to find and to work with because they are ill-suited to standard methods such as coset enumeration.
In particular, the second homology plays a special role and this led Heinz Hopf to find an effective method for calculating it.
The method in (Hopf 1942) is also known as Hopf's integral homology formula and is identical to Schur's formula for the Schur multiplier of a finite group: where
[3] The recognition that these formulas were the same led Samuel Eilenberg and Saunders Mac Lane to the creation of cohomology of groups.
In general, where the star denotes the algebraic dual group.
For one approach and references see the paper by Everaert, Gran and Van der Linden listed below.
The Schur covers of finite perfect groups are superperfect.
The second algebraic K-group K2(R) of a commutative ring R can be identified with the second homology group H2(E(R), Z) of the group E(R) of (infinite) elementary matrices with entries in R.[4] The references from Clair Miller give another view of the Schur Multiplier as the kernel of a morphism κ: G ∧ G → G induced by the commutator map.
