Covering groups of the alternating and symmetric groups
Encyclopedia
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 representation
s of the alternating and symmetric group
s. The covering groups were classified in : for n≥4 the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold.
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.
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 . 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!.
These relations can be used to describe two non-isomorphic covers of the symmetric group. One covering group
has generators z, t1, ..., tn−1 and relations:
The same group
can be given the following presentation using the generators z and si given by ti or tiz according as i is odd or even:
The other covering group
has generators z, t1, ..., tn−1 and relations:
The same group
can be given the following presentation using the generators z and si given by ti or tiz according as i is odd or even:
Sometimes all of the relations of the symmetric group are expressed as (titj)mij = 1, where mij are non-negative integers, namely mii = 1, mi,i+1 = 3, and mij = 2, for 1 ≤ i < i+2 ≤ j ≤ n−1. The presentation of
becomes particularly simple in this form: (titj)mij = z, and zz = 1. 
has the nice property that its generators all have order 2.
to classify projective representation
s 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 .
. The Schur multipliers of the alternating groups An (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 6-fold cover.
For the symmetric group, the Schur multiplier vanishes for n ≤ 3, and is the cyclic group of order 2 for n ≥ 4:
Explicitly, Sn acts on the n-dimensional space Rn by permuting coordinates (in matrices, as permutation matrices). This has a 1-dimensional 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 group
s). The special orthogonal group has a 2-fold cover by the spin group
and restricting this cover to 
and taking the preimage yields a 2-fold cover 
A similar construction with a pin group
yields the 2-fold cover of the symmetric group:
As there are two pin groups, there are two distinct 2-fold covers of the symmetric group, 2⋅Sn±, also called 
and 
.
denoted 
and the corresponding triple cover of 
denoted 
can be constructed as symmetries of a certain set of vector is complex 6-space. While the exceptional triple covers of A6 and A7 extend to extensions
of S6 and S7, these extensions are not central and so do not form Schur covers.
This construction is important in the study of the sporadic group
s, and in much of the exceptional behavior of small classical and exceptional groups, including: construction of the Mathieu group M24, the exceptional covers of the projective unitary group
and the projective special linear group 
and the exceptional double cover of the group of Lie type
s 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) ≅ A4, which can also be thought of as the binary tetrahedral group covering the tetrahedral group. Similarly, GL(2,3) → PGL(2,3) ≅ S4 is a Schur cover, but there is a second non-isomorphic Schur cover of S4 contained in GL(2,9) – note that 9=32 so this is extension of scalars
of GL(2,3). In terms of the above presentations, GL(2,3) ≅ Ŝ4.
For n = 5, the Schur cover of the alternating group is given by SL(2,5) → PSL(2,5) ≅ A5, which can also be thought of as the binary icosahedral group covering the icosahedral group. Though PGL(2,5) ≅ S5, 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=52, so this extends the scalars.
For n = 6, the double cover of the alternating group is given by SL(2,9) → PSL(2,9) ≅ A6. While PGL(2,9) is contained in the automorphism group PΓL(2,9) of PSL(2,9) ≅ A6, PGL(2,9) is not isomorphic to S6, 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 A6, due to the exceptional outer automorphism of A6. Another subgroup of the automorphism group of A6 is M10, the Mathieu group
of degree 10, whose Schur cover is a triple cover. The Schur covers of the symmetric group S6 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 A6 are double covers.
For n = 8, the alternating group A8 is isomorphic to SL(4,2) = PSL(4,2), and so SL(4,2) → PSL(4,2), which is 1-to-1, not 2-to-1, is not a Schur cover.
s are superperfect, that is both their first and second integral homology vanish. In particular, the double covers of An for n ≥ 4 are superperfect, except for n = 6, 7, and the six-fold covers of An are superperfect for n = 6, 7.
As stem extensions of a simple group, the covering groups of An are quasisimple group
s for n ≥ 5.
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representation
Projective representation
In the mathematical field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
s of the alternating and symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
s. The covering groups were classified in : for n≥4 the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold.
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.
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 centerCenter (group theory)In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
and the derived subgroup of D, and - amongst all such homomorphisms, this D has maximal size.
The Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
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 . 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 Sn has a presentation on n−1 generators ti for i = 1, 2, ..., n−1 and relations- titi = 1, for 1 ≤ i ≤ n−1
- ti+1titi+1 = titi+1ti, for 1 ≤ i ≤ n−2
- tjti = titj, for 1 ≤ i < i+2 ≤ j ≤ n−1.
These relations can be used to describe two non-isomorphic covers of the symmetric group. One covering group
has generators z, t1, ..., tn−1 and relations:
- zz = 1
- titi = z, for 1 ≤ i ≤ n−1
- ti+1titi+1 = titi+1ti, for 1 ≤ i ≤ n−2
- tjti = titjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The same group
can be given the following presentation using the generators z and si given by ti or tiz according as i is odd or even:
- zz = 1
- sisi = z, for 1 ≤ i ≤ n−1
- si+1sisi+1 = sisi+1siz, for 1 ≤ i ≤ n−2
- sjsi = sisjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The other covering group
has generators z, t1, ..., tn−1 and relations:
- zz = 1, zti = tiz, for 1 ≤ i ≤ n−1
- titi = 1, for 1 ≤ i ≤ n−1
- ti+1titi+1 = titi+1tiz, for 1 ≤ i ≤ n−2
- tjti = titjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
The same group
can be given the following presentation using the generators z and si given by ti or tiz according as i is odd or even:
- zz = 1, zsi = siz, for 1 ≤ i ≤ n−1
- sisi = 1, for 1 ≤ i ≤ n−1
- si+1sisi+1 = sisi+1si, for 1 ≤ i ≤ n−2
- sjsi = sisjz, for 1 ≤ i < i+2 ≤ j ≤ n−1.
Sometimes all of the relations of the symmetric group are expressed as (titj)mij = 1, where mij are non-negative integers, namely mii = 1, mi,i+1 = 3, and mij = 2, for 1 ≤ i < i+2 ≤ j ≤ n−1. The presentation of
becomes particularly simple in this form: (titj)mij = z, and zz = 1. has the nice property that its generators all have order 2.Projective representations
Covering groups were introduced by Issai SchurIssai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
to classify projective representation
Projective representation
In the mathematical field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
s of groups. A (complex) linear representation of a group G is a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
G → GL(n,C) from the group G to a general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
, while a projective representation is a homomorphism G → PGL(n,C) from G to a projective linear group
Projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P...
. 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 .
Integral homology
Covering groups correspond to the second group homology group, H2(G,Z), also known as the Schur multiplierSchur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
. The Schur multipliers of the alternating groups An (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 6-fold cover.
- H2(An,Z) = 0 for n ≤ 3
- H2(An,Z) = Z/2Z for n = 4, 5
- H2(An,Z) = Z/6Z for n = 6, 7
- H2(An,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:
- H2(Sn,Z) = 0 for n ≤ 3
- H2(Sn,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 An and Sn. These spin representations exist for all n, but are the covering groups only for n≥4 (n≠6,7 for An). For n≤3, Sn and An are their own Schur covers.Explicitly, Sn acts on the n-dimensional space Rn by permuting coordinates (in matrices, as permutation matrices). This has a 1-dimensional trivial subrepresentation
Trivial representation
In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V...
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
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
, and algebraically, it yields maps
and expressing these as discrete subgroups (point groupPoint group
In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
s). The special orthogonal group has a 2-fold cover by the spin group
and restricting this cover to and taking the preimage yields a 2-fold cover A similar construction with a pin groupPin group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group....
yields the 2-fold cover of the symmetric group:
As there are two pin groups, there are two distinct 2-fold covers of the symmetric group, 2⋅Sn±, also called and .Construction of triple cover for n = 6, 7
The triple covering of denoted and the corresponding triple cover of denoted can be constructed as symmetries of a certain set of vector is complex 6-space. While the exceptional triple covers of A6 and A7 extend to extensionsGroup extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
of S6 and S7, these extensions are not central and so do not form Schur covers.
This construction is important in the study of the sporadic group
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
s, and in much of the exceptional behavior of small classical and exceptional groups, including: construction of the Mathieu group M24, the exceptional covers of the projective unitary group
Projective unitary group
In mathematics, the projective unitary group PU is the quotient of the unitary group U by the right multiplication of its center, U, embedded as scalars....
and the projective special linear group and the exceptional double cover of the group of Lie typeGroup of Lie type
In mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
Exceptional isomorphisms
For low dimensions there are exceptional isomorphismExceptional isomorphism
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families of mathematical objects, that is not an example of a pattern of such isomorphisms.Because these series of objects are presented differently, they are not...
s with the map from a special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
over a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
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) ≅ A4, which can also be thought of as the binary tetrahedral group covering the tetrahedral group. Similarly, GL(2,3) → PGL(2,3) ≅ S4 is a Schur cover, but there is a second non-isomorphic Schur cover of S4 contained in GL(2,9) – note that 9=32 so this is extension of scalars
Extension of scalars
In abstract algebra, extension of scalars is a means of producing a module over a ring S from a module over another ring R, given a homomorphism f : R \to S between them...
of GL(2,3). In terms of the above presentations, GL(2,3) ≅ Ŝ4.
For n = 5, the Schur cover of the alternating group is given by SL(2,5) → PSL(2,5) ≅ A5, which can also be thought of as the binary icosahedral group covering the icosahedral group. Though PGL(2,5) ≅ S5, 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=52, so this extends the scalars.
For n = 6, the double cover of the alternating group is given by SL(2,9) → PSL(2,9) ≅ A6. While PGL(2,9) is contained in the automorphism group PΓL(2,9) of PSL(2,9) ≅ A6, PGL(2,9) is not isomorphic to S6, 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 A6, due to the exceptional outer automorphism of A6. Another subgroup of the automorphism group of A6 is M10, the Mathieu groupMathieu group
In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
of degree 10, whose Schur cover is a triple cover. The Schur covers of the symmetric group S6 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 A6 are double covers.
For n = 8, the alternating group A8 is isomorphic to SL(4,2) = PSL(4,2), and so SL(4,2) → PSL(4,2), which is 1-to-1, not 2-to-1, is not a Schur cover.
Properties
Schur covers of finite perfect groupPerfect group
In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
s are superperfect, that is both their first and second integral homology vanish. In particular, the double covers of An for n ≥ 4 are superperfect, except for n = 6, 7, and the six-fold covers of An are superperfect for n = 6, 7.
As stem extensions of a simple group, the covering groups of An are quasisimple group
Quasisimple group
In mathematics, a quasisimple group is a group that is a perfect central extension E of a simple group S...
s for n ≥ 5.