Schur multiplier

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In mathematical

${\displaystyle H_{2}(G,\mathbb {Z} )}$ of a group G. It was introduced by Issai Schur (1904) in his work on projective representations.

The Schur multiplier ${\displaystyle \operatorname {M} (G)}$ of a finite group G is a finite

of G is cyclic for some p, then the order of ${\displaystyle \operatorname {M} (G)}$ is not divisible by p. In particular, if all of G are cyclic, then ${\displaystyle \operatorname {M} (G)}$ is trivial.

For instance, the Schur multiplier of the

has order 2.

The Schur multipliers of the finite simple groups are given at the list of finite simple groups. The covering groups of the alternating and symmetric groups are of considerable recent interest.

central extension

C of G.

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 ${\displaystyle H^{2}(G,\mathbb {C} ^{\times })}$. A projective representation is much like a group representation except that instead of a homomorphism into the general linear group ${\displaystyle \operatorname {GL} (n,\mathbb {C} )}$, one takes a homomorphism into the

${\displaystyle \operatorname {PGL} (n,\mathbb {C} )}$. In other words, a projective representation is a representation modulo the .

.

The study of such covering groups led naturally to the study of

and stem extensions.

A

of a group G is an extension

${\displaystyle 1\to K\to C\to G\to 1}$

where ${\displaystyle K\leq Z(C)}$ is a subgroup of the center of C.

A stem extension of a group G is an extension

${\displaystyle 1\to K\to C\to G\to 1}$

where ${\displaystyle K\leq Z(C)\cap C'}$ is a subgroup of the intersection of the center of C and the

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

isoclinic

.

It is also called more briefly a universal central extension, but note that there is no largest central extension, as the direct product of G and an abelian group form a central extension of G of arbitrary size.

Stem extensions have the nice property that any lift of a generating set of G is a generating set of C. If the group G is presented in terms of a free group F on a set of generators, and a normal subgroup R generated by a set of relations on the generators, so that ${\displaystyle G\cong F/R}$, then the covering group itself can be presented in terms of F but with a smaller normal subgroup S, that is, ${\displaystyle C\cong F/S}$. Since the relations of G specify elements of K when considered as part of C, one must have ${\displaystyle S\leq [F,R]}$.

In fact if G is perfect, this is all that is needed: C ≅ [F,F]/[F,R] and M(G) ≅ K ≅ R/[F,R]. Because of this simplicity, expositions such as (Aschbacher 2000, §33) handle the perfect case first. The general case for the Schur multiplier is similar but ensures the extension is a stem extension by restricting to the derived subgroup of F: M(G) ≅ (R ∩ [F, F])/[F, R]. These are all slightly later results of Schur, who also gave a number of useful criteria for calculating them more explicitly.

In

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 topology, groups can often be described as finitely presented groups and a fundamental question is to calculate their integral homology ${\displaystyle H_{n}(G,\mathbb {Z} )}$. 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:

${\displaystyle H_{2}(G,\mathbb {Z} )\cong (R\cap [F,F])/[F,R]}$

where ${\displaystyle G\cong F/R}$ and F is a free group. The same formula also holds when G is a perfect group.^[3]

The recognition that these formulas were the same led

. In general,

${\displaystyle H_{2}(G,\mathbb {Z} )\cong {\bigl (}H^{2}(G,\mathbb {C} ^{\times }){\bigr )}^{*}}$

where the star denotes the algebraic dual group. Moreover, when G is finite, there is an unnatural isomorphism

${\displaystyle {\bigl (}H^{2}(G,\mathbb {C} ^{\times }){\bigr )}^{*}\cong H^{2}(G,\mathbb {C} ^{\times }).}$

The Hopf formula for ${\displaystyle H_{2}(G)}$ has been generalised to higher dimensions. For one approach and references see the paper by Everaert, Gran and Van der Linden listed below.

A

is a group all of whose reduced integral homology vanishes.

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]

• Quasisimple group

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.