Schur multiplier

From Wikipedia, the free encyclopedia

In mathematics, more specifically in group theory, the Schur multiplier is an important invariant of a group that has applications in many areas of mathematics. The Schur multiplier is sometimes called the multiplicator and is named after Issai Schur. It is now usually defined as the second homology group of a group G with coefficients in the integers, $H_2(G, {\Bbb Z})$ .

1 Relation to projective representations
2 Relation to central extensions
- 2.1 Properties
3 Relation to efficient presentations
4 Relation to topology
5 See also
6 References

[edit] Relation to projective representations

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

One of Schur's achievements is to show 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 covers of the finite simple groups are known, and each is an example of a quasisimple group.

[edit] Relation to central extensions

The study of such covering groups led naturally to the study of central and stem extensions.

A central extension of a group G is an extension

$1 \to K \to C \to G \to 1$

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

A stem extension of a group G is an extension

$1 \to K \to C \to G \to 1$

where $K \leq Z(C) \cap C'$ is a subgroup of the intersection of the center of C and the derived subgroup of C; this is more restrictive than central.

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).

It is also called more briefly a universal central extension, but note that there is no largest central extension, as $C_n \times G$ is a central extension of $G$ of arbitrary size.

[edit] Properties

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 $G \cong F/R$ , then the covering group itself can be presented in terms of F but with a smaller normal subgroup S, $C \cong F/S$ . Since the relations of G specify elements of K when considered as part of C, one must have $S \leq [F,R]$ .

In fact if G is perfect, this is all that is needed: $C \cong [F,F]/[F,R]$ and $M(G) \cong K \cong R/[F,R]$ . Because of this simplicity, many expositions 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: $M(G) \cong (R \cap [F, F])/[F, R]$ . These are all slightly later results of Schur, who also gave a number of useful criteria for calculating them more explicitly.

[edit] Relation to efficient presentations

In many areas of mathematics, a group almost always originates from a presentation. With such strong stimulus, the study of presentations themselves developed into an important branch of group theory called combinatorial group theory. One important theme in this area of mathematics is to study presentations with as few relations as possible, such as one relator groups like Baumslag-Solitar groups. These groups are infinite groups with two generators and one relation, and an old result of Schreier shows that in any presentation with more generators than relations, the resulting group is infinite. The borderline case is thus quite interesting: finite groups with the same number of generators as relations are said to have an efficient presentation. For a group to have an efficient presentation, 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.

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.

[edit] Relation to topology

In topology, groups quite often originate as finitely presented groups and a fundamental question is to calculate their integral homology $H_n(G,{\Bbb Z})$ . In particular, the second homology plays a special role and this led Hopf to find effective methods for calculating it. His 1942 result^[1] is now known as Hopf's integral homology formula and is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group:

$H_2(G,{\Bbb Z}) \cong (R \cap [F, F])/[F, R]$ where $G \cong F/R$ and F is a free group.

The recognition that these formulas were the same has been said to have led Eilenberg and Mac Lane to the creation of cohomology of groups. In general, $H_2(G,{\Bbb Z}) \cong \bigl( H^2(G,{\Bbb C}^\times) \bigr)^*$ where the star denotes the algebraic dual group, and when G is finite, there is an unnatural isomorphism $\bigl( H^2(G,{\Bbb C}^\times) \bigr)^* \cong H^2(G,{\Bbb C}^\times)$ .

[edit] See also

[edit] References

Aschbacher, Michael: Finite Group Theory, Cambridge University Press, 2000, ISBN 0-521-78675-4
Karpilovsky, Gregory: The Schur Multiplier, Oxford University Press, 1987, ISBN 0-19-853554-6.
Rotman, Joseph: An Introduction to the Theory of Groups, Springer Verlag, 1995, ISBN 0-387-94285-8
Wiegold, J.: The Schur multiplier: an elementary approach, in Groups, St Andrews, 1981, London Mathematical Society Lecture Note Series 71 (Cambridge University Press, Cambridge, 1982) 137–154

^ Hopf, Heinz. "Fundamentalgruppe und zweite Bettische Gruppe." Comment. Math. Helv. 14, (1942). 257--309. MR 6510

Categories: Group theory | Homological algebra

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Schur multiplier

From Wikipedia, the free encyclopedia

Contents

[edit] Relation to projective representations

[edit] Relation to central extensions

[edit] Properties

[edit] Relation to efficient presentations

[edit] Relation to topology

[edit] See also

[edit] References

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