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In the mathematical area of
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
s. The covering groups were classified in : for , 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 center and the
commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
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 . The symmetric group of degree has Schur covers of order 2⋅''n''! There are two isomorphism classes if and one isomorphism class if ''n'' = 6. 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 by means of generators and relations. The symmetric group S''n'' has a
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
on generators ''t''''i'' for ''i'' = 1, 2, ..., and relations : ''t''''i''''t''''i'' = 1, for : ''t''''i''+1''t''''i''''t''''i''+1 = ''t''''i''''t''''i''+1''t''''i'', for : ''t''''j''''t''''i'' = ''t''''i''''t''''j'', for . These relations can be used to describe two non-isomorphic covers of the symmetric group. One covering group 2⋅S has generators ''z'', ''t''1, ..., ''t''''n''−1 and relations: : ''zz'' = 1 : ''t''''i''''t''''i'' = ''z'', for : ''t''''i''+1''t''''i''''t''''i''+1 = ''t''''i''''t''''i''+1''t''''i'', : ''t''''j''''t''''i'' = ''t''''i''''t''''j''''z'', for . The same group 2⋅S 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 : ''s''''i''+1''s''''i''''s''''i''+1 = ''s''''i''''s''''i''+1''s''''i''''z'', for : ''s''''j''''s''''i'' = ''s''''i''''s''''j''''z'', . The other covering group 2⋅S has generators ''z'', ''t''1, ..., ''t''''n''−1 and relations: : ''zz'' = 1, ''zt''''i'' = ''t''''i''''z'', for : ''t''''i''''t''''i'' = 1, for : ''t''''i''+1''t''''i''''t''''i''+1 = ''t''''i''''t''''i''+1''t''''i''''z'', for : ''t''''j''''t''''i'' = ''t''''i''''t''''j''''z'', for . The same group 2⋅S 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 : ''s''''i''''s''''i'' = 1, for : ''s''''i''+1''s''''i''''s''''i''+1 = ''s''''i''''s''''i''+1''s''''i'', for : ''s''''j''''s''''i'' = ''s''''i''''s''''j''''z'', for . Sometimes all of the relations of the symmetric group are expressed as , where ''m''''ij'' are non-negative integers, namely , , and , for . The presentation of 2⋅S becomes particularly simple in this form: () = ''z'', and ''zz'' = 1. The group 2⋅S has the nice property that its generators all have order 2.


Projective representations

Covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous (topology), continuous group homomorphism. The map ''p'' is called the c ...
s were introduced by
Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...
to classify projective representations of groups. A (complex) ''linear'' representation of a group ''G'' is a
group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...
from the group ''G'' to a
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
, while a ''projective'' representation is a homomorphism from ''G'' to a
projective linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
. 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, , 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 6-fold cover. : H2(A''n'', Z) = 0 for ''n'' ≤ 3 : H2(A''n'', Z) = Z/2Z for ''n'' = 4, 5 : H2(A''n'', Z) = Z/6Z for ''n'' = 6, 7 : H2(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: : H2(S''n'', Z) = 0 for ''n'' ≤ 3 : H2(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 ''n''-dimensional space R''n'' by permuting coordinates (in matrices, as
permutation matrices In mathematics, particularly in Matrix (mathematics), matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permu ...
). This has a 1-dimensional trivial subrepresentation corresponding to vectors with all coordinates equal, and the complementary -dimensional subrepresentation (of vectors whose coordinates sum to 0) is irreducible for . Geometrically, this is the symmetries of the - simplex, and algebraically, it yields maps A_n \hookrightarrow \operatorname(n-1) and S_n \hookrightarrow \operatorname(n-1) expressing these as discrete subgroups (
point group In geometry, a point group is a group (mathematics), mathematical group of symmetry operations (isometry, isometries in a Euclidean space) that have a Fixed point (mathematics), fixed point in common. The Origin (mathematics), coordinate origin o ...
s). The special orthogonal group has a 2-fold cover by the spin group , and restricting this cover to A''n'' 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⋅S, also called \tilde \mathrm_n and Ŝ''n''.


Construction of triple cover for ''n'' = 6, 7

The triple covering of A6, denoted 3⋅A6, and the corresponding triple cover of S6, denoted 3⋅S6, can be constructed as symmetries of a certain set of vectors in a 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 groups, 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 ''U''4(3) and the projective special linear group L_3(4), and the exceptional double cover of the
group of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a Reductive group, reductive linear algebraic group with values in a finite ...
G2(4).


Exceptional isomorphisms

For low dimensions there are exceptional isomorphisms with the map from a special linear group over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
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, S_n \cong \operatorname(A_n), 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.


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 six-fold 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

* * ** * {{refend Finite groups Permutation groups