 Covering groups of the alternating and symmetric groups

In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classified in (Schur 1911): for n≥4 the covering groups are 2fold covers except for the alternating groups of degree 6 and 7 where the covers are 6fold.
For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4.
Contents
Definition and classification
A group homomorphism from D to G is said to be a Schur cover of the finite group G if:
 the kernel is contained both in the center and the derived subgroup of D, and
 amongst all such homomorphisms, this D has maximal size.
The Schur multiplier of G is the kernel of any Schur cover and has many interpretations. When the homomorphism is understood, the group D is often called the Schur cover or Darstellungsgruppe.
The Schur covers of the symmetric and alternating groups were classified in (Schur 1911). The symmetric group of degree n ≥ 4 has two isomorphism classes of Schur covers, both of order 2⋅n!, and the alternating group of degree n has one isomorphism class of Schur cover, which has order n! except when n is 6 or 7, in which case the Schur cover has order 3⋅n!.
Finite presentations
Schur covers can be described using finite presentations. The symmetric group S_{n} has a presentation on n−1 generators t_{i} for i = 1, 2, ..., n−1 and relations
 t_{i}t_{i} = 1, for 1 ≤ i ≤ n−1
 t_{i+1}t_{i}t_{i+1} = t_{i}t_{i+1}t_{i}, for 1 ≤ i ≤ n−2
 t_{j}t_{i} = t_{i}t_{j}, for 1 ≤ i < i+2 ≤ j ≤ n−1.
These relations can be used to describe two nonisomorphic covers of the symmetric group. One covering group has generators z, t_{1}, ..., t_{n−1} and relations:
 zz = 1
 t_{i}t_{i} = z, for 1 ≤ i ≤ n−1
 t_{i+1}t_{i}t_{i+1} = t_{i}t_{i+1}t_{i}, for 1 ≤ i ≤ n−2
 t_{j}t_{i} = t_{i}t_{j}z, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The same group can be given the following presentation using the generators z and s_{i} given by t_{i} or t_{i}z according as i is odd or even:
 zz = 1
 s_{i}s_{i} = z, for 1 ≤ i ≤ n−1
 s_{i+1}s_{i}s_{i+1} = s_{i}s_{i+1}s_{i}z, for 1 ≤ i ≤ n−2
 s_{j}s_{i} = s_{i}s_{j}z, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The other covering group has generators z, t_{1}, ..., t_{n−1} and relations:
 zz = 1, zt_{i} = t_{i}z, for 1 ≤ i ≤ n−1
 t_{i}t_{i} = 1, for 1 ≤ i ≤ n−1
 t_{i+1}t_{i}t_{i+1} = t_{i}t_{i+1}t_{i}z, for 1 ≤ i ≤ n−2
 t_{j}t_{i} = t_{i}t_{j}z, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The same group can be given the following presentation using the generators z and s_{i} given by t_{i} or t_{i}z according as i is odd or even:
 zz = 1, zs_{i} = s_{i}z, for 1 ≤ i ≤ n−1
 s_{i}s_{i} = 1, for 1 ≤ i ≤ n−1
 s_{i+1}s_{i}s_{i+1} = s_{i}s_{i+1}s_{i}, for 1 ≤ i ≤ n−2
 s_{j}s_{i} = s_{i}s_{j}z, for 1 ≤ i < i+2 ≤ j ≤ n−1.
Sometimes all of the relations of the symmetric group are expressed as (t_{i}t_{j})^{mij} = 1, where m_{ij} are nonnegative integers, namely m_{ii} = 1, m_{i,i+1} = 3, and m_{ij} = 2, for 1 ≤ i < i+2 ≤ j ≤ n−1. The presentation of becomes particularly simple in this form: (t_{i}t_{j})^{mij} = z, and zz = 1. has the nice property that its generators all have order 2.
Projective representations
Covering groups were introduced by Issai Schur to classify projective representations of groups. A (complex) linear representation of a group G is a group homomorphism G → GL(n,C) from the group G to a general linear group, while a projective representation is a homomorphism G → PGL(n,C) from G to a projective linear group. Projective representations of G correspond naturally to linear representations of the covering group of G.
The projective representations of alternating and symmetric groups are the subject of the book (Hoffman & Humphreys 1992).
Integral homology
Covering groups correspond to the second group homology group, H_{2}(G,Z), also known as the Schur multiplier. The Schur multipliers of the alternating groups A_{n} (in the case where n is at least 4) are the cyclic groups of order 2, except in the case where n is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is the cyclic group of order 6, and the covering group is a 6fold cover.
 H_{2}(A_{n},Z) = 0 for n ≤ 3
 H_{2}(A_{n},Z) = Z/2Z for n = 4, 5
 H_{2}(A_{n},Z) = Z/6Z for n = 6, 7
 H_{2}(A_{n},Z) = Z/2Z for n ≥ 8
For the symmetric group, the Schur multiplier vanishes for n ≤ 3, and is the cyclic group of order 2 for n ≥ 4:
 H_{2}(S_{n},Z) = 0 for n ≤ 3
 H_{2}(S_{n},Z) = Z/2Z for n ≥ 4
Construction of double covers
The double covers can be constructed as spin (respectively, pin) covers of faithful, irreducible, linear representations of A_{n} and S_{n}. These spin representations exist for all n, but are the covering groups only for n≥4 (n≠6,7 for A_{n}). For n≤3, S_{n} and A_{n} are their own Schur covers.
Explicitly, S_{n} acts on the ndimensional space R^{n} by permuting coordinates (in matrices, as permutation matrices). This has a 1dimensional trivial subrepresentation corresponding to vectors with all coordinates equal, and the complementary (n−1)dimensional subrepresentation (of vectors whose coordinates sum to 0) is irreducible for n≥4. Geometrically, this is the symmetries of the (n−1)simplex, and algebraically, it yields maps and expressing these as discrete subgroups (point groups). The special orthogonal group has a 2fold cover by the spin group and restricting this cover to A_{n} and taking the preimage yields a 2fold cover A similar construction with a pin group yields the 2fold cover of the symmetric group: As there are two pin groups, there are two distinct 2fold covers of the symmetric group, 2⋅S_{n}^{±}, also called and .
Construction of triple cover for n = 6, 7
Main article: Valentiner groupThe triple covering of A_{6}, denoted and the corresponding triple cover of S_{6}, denoted can be constructed as symmetries of a certain set of vector is complex 6space. While the exceptional triple covers of A_{6} and A_{7} extend to extensions of S_{6} and S_{7}, these extensions are not central and so do not form Schur covers.
This construction is important in the study of the sporadic groups, and in much of the exceptional behavior of small classical and exceptional groups, including: construction of the Mathieu group M_{24}, the exceptional covers of the projective unitary group U_{4}(3) and the projective special linear group L_{3}(4), and the exceptional double cover of the group of Lie type G_{2}(4).
Exceptional isomorphisms
For low dimensions there are exceptional isomorphisms with the map from a special linear group over a finite field to the projective special linear group.
For n = 3, the symmetric group is SL(2,2) ≅ PSL(2,2) and is its own Schur cover.
For n = 4, the Schur cover of the alternating group is given by SL(2,3) → PSL(2,3) ≅ A_{4}, which can also be thought of as the binary tetrahedral group covering the tetrahedral group. Similarly, GL(2,3) → PGL(2,3) ≅ S_{4} is a Schur cover, but there is a second nonisomorphic Schur cover of S_{4} contained in GL(2,9) – note that 9=3^{2} so this is extension of scalars of GL(2,3). In terms of the above presentations, GL(2,3) ≅ Ŝ_{4}.
For n = 5, the Schur cover of the alternating group is given by SL(2,5) → PSL(2,5) ≅ A_{5}, which can also be thought of as the binary icosahedral group covering the icosahedral group. Though PGL(2,5) ≅ S_{5}, GL(2,5) → PGL(2,5) is not a Schur cover as the kernel is not contained in the derived subgroup of GL(2,5). The Schur cover of PGL(2,5) is contained in GL(2,25) – as before, 25=5^{2}, so this extends the scalars.
For n = 6, the double cover of the alternating group is given by SL(2,9) → PSL(2,9) ≅ A_{6}. While PGL(2,9) is contained in the automorphism group PΓL(2,9) of PSL(2,9) ≅ A_{6}, PGL(2,9) is not isomorphic to S_{6}, and its Schur covers (which are double covers) are not contained in nor a quotient of GL(2,9). Note that in almost all cases, with the unique exception of A_{6}, due to the exceptional outer automorphism of A_{6}. Another subgroup of the automorphism group of A_{6} is M_{10}, the Mathieu group of degree 10, whose Schur cover is a triple cover. The Schur covers of the symmetric group S_{6} itself have no faithful representations as a subgroup of GL(d,9) for d≤3. The four Schur covers of the automorphism group PΓL(2,9) of A_{6} are double covers.
For n = 8, the alternating group A_{8} is isomorphic to SL(4,2) = PSL(4,2), and so SL(4,2) → PSL(4,2), which is 1to1, not 2to1, is not a Schur cover.
Properties
Schur covers of finite perfect groups are superperfect, that is both their first and second integral homology vanish. In particular, the double covers of A_{n} for n ≥ 4 are superperfect, except for n = 6, 7, and the sixfold covers of A_{n} are superperfect for n = 6, 7.
As stem extensions of a simple group, the covering groups of A_{n} are quasisimple groups for n ≥ 5.
References
 Hoffman, P. N.; Humphreys, John F. (1992), Projective representations of the symmetric groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 9780198535560, MR1205350
 Schur, J. (1911), "Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen", Journal für die reine und angewandte Mathematik 139: 155–250, JFM 42.0154.02, http://www.digizeitschriften.de/resolveppn/GDZPPN002167298
 Schur, J. (2001), "On the representation of the symmetric and alternating groups by fractional linear substitutions", International Journal of Theoretical Physics 40 (1): 413–458, doi:10.1023/A:1003772419522, ISSN 00207748, MR1820589, Zbl 0969.20002(translated from the German)
 Wilson, Robert (October 31, 2006), "Chapter 2: Alternating groups", http://www.maths.qmul.ac.uk/~raw/fsgs_files/alt.ps, 2.7: Covering groups
Categories: Finite groups
 Permutation groups
Wikimedia Foundation. 2010.
Look at other dictionaries:
Classification of finite simple groups — Group theory Group theory … Wikipedia
List of finite simple groups — In mathematics, the classification of finite simple groups states thatevery finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type (including the Tits group, which strictly speaking is not of Lie type),or… … Wikipedia
Schur multiplier — In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by Issai Schur (1904) in his work on projective representations. Contents 1 Examples and properties 2 Re … Wikipedia
Lorentz group — Group theory Group theory … Wikipedia
Orthogonal group — Group theory Group theory … Wikipedia
Orthogonal matrix — In linear algebra, an orthogonal matrix (less commonly called orthonormal matrix[1]), is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Equivalently, a matrix Q is orthogonal if… … Wikipedia
Rotation matrix — In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian… … Wikipedia
MUSIC — This article is arranged according to the following outline: introduction written sources of direct and circumstantial evidence the material relics and iconography notated sources oral tradition archives and important collections of jewish music… … Encyclopedia of Judaism
Projective linear group — In mathematics, especially in area of algebra called group theory, the projective linear group (also known as the projective general linear group) is one of the fundamental groups of study, part of the so called classical groups. The projective… … Wikipedia
Ericales — ▪ plant order Introduction rhododendron order of flowering plants, containing 25 families, 346 genera, and more than 11,000 species. The relationships of the order are unclear. It belongs to neither of the two major asterid groups… … Universalium