View source for 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 representation]]s of the [[alternating group|alternating]] and [[symmetric group]]s.  The covering groups were classified in {{Harv|Schur|1911}}: for {{nowrap|''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 [[center (group theory)|center]] and the [[commutator 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 {{harv|Schur|1911}}.  The symmetric group of degree {{nowrap|''n'' ≥ 4}} has 
Schur covers of order 2⋅''n''! There are two isomorphism classes if {{nowrap|''n'' ≠ 6}} 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<sub>''n''</sub> has a [[Presentation of a group|presentation]] on {{nowrap|''n'' − 1}} generators ''t''<sub>''i''</sub> for ''i'' = 1, 2, ..., {{nowrap|n − 1}} and relations
: ''t''<sub>''i''</sub>''t''<sub>''i''</sub> = 1, for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}
: ''t''<sub>''i''+1</sub>''t''<sub>''i''</sub>''t''<sub>''i''+1</sub> = ''t''<sub>''i''</sub>''t''<sub>''i''+1</sub>''t''<sub>''i''</sub>, for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 2}}
: ''t''<sub>''j''</sub>''t''<sub>''i''</sub> = ''t''<sub>''i''</sub>''t''<sub>''j''</sub>, for {{nowrap|1 ≤ ''i'' &lt; ''i'' + 2 ≤ ''j'' ≤ ''n'' − 1}}.

These relations can be used to describe two non-isomorphic covers of the symmetric group.  One covering group 2⋅S{{su|lh=1|b=''n''|p=−}} has generators ''z'', ''t''<sub>1</sub>, ..., ''t''<sub>''n''−1</sub> and relations:
: ''zz'' = 1
: ''t''<sub>''i''</sub>''t''<sub>''i''</sub> = ''z'', for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}
: ''t''<sub>''i''+1</sub>''t''<sub>''i''</sub>''t''<sub>''i''+1</sub> = ''t''<sub>''i''</sub>''t''<sub>''i''+1</sub>''t''<sub>''i''</sub>, {{nowrap|for 1 ≤ ''i'' ≤ ''n'' − 2}}
: ''t''<sub>''j''</sub>''t''<sub>''i''</sub> = ''t''<sub>''i''</sub>''t''<sub>''j''</sub>''z'', for {{nowrap|1 ≤ ''i'' &lt; ''i'' + 2 ≤ ''j'' ≤ ''n'' − 1}}.
The same group 2⋅S{{su|lh=1|b=''n''|p=−}} can be given the following presentation using the generators ''z'' and ''s''<sub>''i''</sub> given by ''t''<sub>''i''</sub> or ''t''<sub>''i''</sub>''z'' according as ''i'' is odd or even:
: ''zz'' = 1
: ''s''<sub>''i''</sub>''s''<sub>''i''</sub> = ''z'', for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}
: ''s''<sub>''i''+1</sub>''s''<sub>''i''</sub>''s''<sub>''i''+1</sub> = ''s''<sub>''i''</sub>''s''<sub>''i''+1</sub>''s''<sub>''i''</sub>''z'', for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 2}}
: ''s''<sub>''j''</sub>''s''<sub>''i''</sub> = ''s''<sub>''i''</sub>''s''<sub>''j''</sub>''z'', {{nowrap|for 1 ≤ ''i'' &lt; ''i'' + 2 ≤ ''j'' ≤ ''n'' − 1}}.

The other covering group 2⋅S{{su|lh=1|b=''n''|p=+}} has generators ''z'', ''t''<sub>1</sub>, ..., ''t''<sub>''n''−1</sub> and relations:
: ''zz'' = 1, ''zt''<sub>''i''</sub> = ''t''<sub>''i''</sub>''z'', for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}
: ''t''<sub>''i''</sub>''t''<sub>''i''</sub> = 1, for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}
: ''t''<sub>''i''+1</sub>''t''<sub>''i''</sub>''t''<sub>''i''+1</sub> = ''t''<sub>''i''</sub>''t''<sub>''i''+1</sub>''t''<sub>''i''</sub>''z'', for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 2}}
: ''t''<sub>''j''</sub>''t''<sub>''i''</sub> = ''t''<sub>''i''</sub>''t''<sub>''j''</sub>''z'', for {{nowrap|1 ≤ ''i'' &lt; ''i''+2 ≤ ''j'' ≤ ''n'' − 1}}.
The same group 2⋅S{{su|lh=1|b=''n''|p=+}} can be given the following presentation using the generators ''z'' and ''s''<sub>''i''</sub> given by ''t''<sub>''i''</sub> or ''t''<sub>''i''</sub>''z'' according as ''i'' is odd or even:
: ''zz'' = 1, ''zs''<sub>''i''</sub> = ''s''<sub>''i''</sub>''z'', for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}
: ''s''<sub>''i''</sub>''s''<sub>''i''</sub> = 1, for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}
: ''s''<sub>''i''+1</sub>''s''<sub>''i''</sub>''s''<sub>''i''+1</sub> = ''s''<sub>''i''</sub>''s''<sub>''i''+1</sub>''s''<sub>''i''</sub>, for {{nowrap|1 ≤ ''i'' ≤ ''n'' − 2}}
: ''s''<sub>''j''</sub>''s''<sub>''i''</sub> = ''s''<sub>''i''</sub>''s''<sub>''j''</sub>''z'', for {{nowrap|1 ≤ ''i'' &lt; ''i'' + 2 ≤ ''j'' ≤ ''n'' − 1}}.

Sometimes all of the relations of the symmetric group are expressed as {{nowrap|1=({{not a typo|''t''<sub>''i''</sub>''t''<sub>''j''</sub>}})<sup>{{not a typo|''m''<sub>''ij''</sub>}}</sup> = 1}}, where ''m''<sub>''ij''</sub> are non-negative integers, namely {{nowrap|1=''m''<sub>''ii''</sub> = 1}}, {{nowrap|1=''m''<sub>''i'',''i''+1</sub> = 3}}, and {{nowrap|1=''m''<sub>''ij''</sub> = 2}}, for {{nowrap|1 ≤ ''i'' &lt; ''i'' + 2 ≤ ''j'' ≤ ''n'' − 1}}.  The presentation of 2⋅S{{su|lh=1|b=''n''|p=−}} becomes particularly simple in this form: ({{not a typo|''t''<sub>''i''</sub>''t''<sub>''j''</sub>}})<sup>{{not a typo|''m''<sub>''ij''</sub>}}</sup> = ''z'', and ''zz'' = 1.  The group 2⋅S{{su|lh=1|b=''n''|p=+}} has the nice property that its generators all have order 2.

== Projective representations ==

[[Covering group]]s were introduced by [[Issai Schur]] to classify [[projective representation]]s of groups. A (complex) [[linear representation|''linear'' representation]] of a group ''G'' is a [[group homomorphism]] {{nowrap|''G'' → GL(''n'', ''C'')}} from the group ''G'' to a [[general linear group]], while a ''projective'' representation is a homomorphism {{nowrap|''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 {{harv|Hoffman|Humphreys|1992}}.

== Integral homology ==
Covering groups correspond to the second [[group homology]] group, {{not a typo|H<sub>2</sub>(''G'', '''Z''')}}, also known as the [[Schur multiplier]]. The Schur multipliers of the alternating groups A<sub>''n''</sub> (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.
: H<sub>2</sub>(A<sub>''n''</sub>, '''Z''') = 0 for ''n'' ≤ 3
: H<sub>2</sub>(A<sub>''n''</sub>, '''Z''') = '''Z'''/2'''Z''' for ''n'' = 4, 5
: H<sub>2</sub>(A<sub>''n''</sub>, '''Z''') = '''Z'''/6'''Z''' for ''n'' = 6, 7
: H<sub>2</sub>(A<sub>''n''</sub>, '''Z''') = '''Z'''/2'''Z''' 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<sub>2</sub>(S<sub>''n''</sub>, '''Z''') = 0 for ''n'' ≤ 3
: H<sub>2</sub>(S<sub>''n''</sub>, '''Z''') = '''Z'''/2'''Z''' for ''n'' ≥ 4

== Construction of double covers ==
[[File:An-spin-representation.svg|thumb|331px|The double cover of the alternating group can be constructed via the [[spin representation]] that covers the usual linear representation of the alternating group.]]
The double covers can be constructed as spin (respectively, pin) covers of faithful, irreducible, linear representations of A<sub>''n''</sub> and S<sub>''n''</sub>. These spin representations exist for all ''n,'' but are the covering groups only for ''n'' ≥ 4 (''n'' ≠ 6, 7 for A<sub>''n''</sub>).  For ''n'' ≤ 3, S<sub>''n''</sub> and A<sub>''n''</sub> are their own Schur covers.

[[File:An-Sn-covering-groups.svg|thumb|360px|The alternating group, symmetric group, and their double covers are related in this way, and have orthogonal representations and covering spin/pin representations in [[:File:Spin-Pin-SO-O-definite.svg|the corresponding diagram of orthogonal and spin/pin groups]].]]
Explicitly, S<sub>''n''</sub> acts on the ''n''-dimensional space '''R'''<sup>''n''</sup> by permuting coordinates (in matrices, as [[permutation matrices]]). This has a 1-dimensional [[trivial representation|trivial subrepresentation]] corresponding to vectors with all coordinates equal, and the complementary {{nowrap|(''n'' − 1)}}-dimensional subrepresentation (of vectors whose coordinates sum to 0) is irreducible for {{nowrap|''n'' ≥ 4}}. Geometrically, this is the symmetries of the {{nowrap|(''n'' − 1)}}-[[simplex]], and algebraically, it yields maps <math>A_n \hookrightarrow \operatorname{SO}(n-1)</math> and <math>S_n \hookrightarrow \operatorname{O}(n-1)</math> expressing these as discrete subgroups ([[point group]]s). The special orthogonal group has a 2-fold cover by the [[spin group]] {{nowrap|Spin(''n'') → SO(''n'')}}, and restricting this cover to A<sub>''n''</sub> and taking the preimage yields a 2-fold cover {{nowrap|2⋅A<sub>''n''</sub> → A<sub>''n''</sub>}}. A similar construction with a [[pin group]] yields the 2-fold cover of the symmetric group: {{nowrap|Pin<sub>±</sub>(''n'') → O(''n'')}}. As there are two pin groups, there are two distinct 2-fold covers of the symmetric group, 2⋅S{{su|lh=1|b=''n''|p=±}}, also called <math>\tilde \mathrm{S}_n</math> and Ŝ<sub>''n''</sub>.

== Construction of triple cover for ''n''&nbsp;=&nbsp;6, 7 ==
{{main|Valentiner group}}
The triple covering of A<sub>6</sub>, denoted 3⋅A<sub>6</sub>, and the corresponding triple cover of S<sub>6</sub>, denoted 3⋅S<sub>6</sub>, can be constructed as symmetries of a certain set of vectors in a complex 6-space. While the exceptional triple covers of A<sub>6</sub> and A<sub>7</sub> extend to [[group extension|extensions]] of S<sub>6</sub> and S<sub>7</sub>, these extensions are not [[central extension (mathematics)|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 M<sub>24</sub>, the exceptional covers of the [[projective unitary group]] ''U''<sub>4</sub>(3) and the [[projective special linear group]] <math>L_3(4),</math> and the exceptional double cover of the [[group of Lie type]] G<sub>2</sub>(4).{{citation needed|date=October 2021|reason=given ref was not a reliable source, not even a proofread draft}}

== Exceptional isomorphisms ==
For low dimensions there are [[exceptional isomorphism]]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) ≅ A<sub>4</sub>, which can also be thought of as the [[binary tetrahedral group]] covering the [[tetrahedral group]].  Similarly, GL(2, 3) → PGL(2, 3) ≅ S<sub>4</sub> is a Schur cover, but there is a second non-isomorphic Schur cover of S<sub>4</sub> contained in GL(2,9) – note that 9 = 3<sup>2</sup> so this is [[extension of scalars]] of GL(2, 3).  In terms of the above presentations, GL(2, 3) ≅ Ŝ<sub>4</sub><!-- , and a choice of generators are ''t''<sub>1</sub> = [1, 0; 1, −1], ''t''<sub>2</sub> = [1, 0; 0, −1], and ''t''<sub>3</sub> = [1, 1; 0, −1] -->.

For ''n'' = 5, the Schur cover of the alternating group is given by SL(2, 5) → PSL(2, 5) ≅ A<sub>5</sub>, which can also be thought of as the [[binary icosahedral group]] covering the [[icosahedral group]].  Though PGL(2, 5) ≅ S<sub>5</sub>, 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<sup>2</sup>, 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<sub>6</sub>.  While PGL(2, 9) is contained in the automorphism group [[PΓL]](2, 9) of PSL(2, 9) ≅ A<sub>6</sub>, PGL(2, 9) is not isomorphic to S<sub>6</sub>, 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, <math>S_n \cong \operatorname{Aut}(A_n),</math> with the unique exception of A<sub>6</sub>, due to [[The exceptional outer automorphism of S6|the exceptional outer automorphism of A<sub>6</sub>]]. Another subgroup of the automorphism group of A<sub>6</sub> is M<sub>10</sub>, the [[Mathieu group]] of degree 10, whose Schur cover is a triple cover.  The Schur covers of the symmetric group S<sub>6</sub> 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<sub>6</sub> are double covers.

For ''n'' = 8, the alternating group A<sub>8</sub> 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 group]]s are [[superperfect group|superperfect]], that is both their first and second integral homology vanish.  In particular, the double covers of A<sub>''n''</sub> for ''n'' ≥ 4 are superperfect, except for ''n'' = 6, 7, and the six-fold covers of A<sub>''n''</sub> are superperfect for ''n'' = 6, 7.

As stem extensions of a simple group, the covering groups of A<sub>''n''</sub> are [[quasisimple group]]s for ''n'' ≥ 5.

== References ==
{{reflist}}
{{refbegin}}
* {{citation | last1=Hoffman | first1=P. N. | last2=Humphreys | first2=John F. | title=Projective representations of the symmetric groups | publisher=The Clarendon Press Oxford University Press | series=Oxford Mathematical Monographs | isbn=978-0-19-853556-0 | mr=1205350 | year=1992}}
* {{citation
|first1=J.
|last1=Schur
|author1-link=Issai Schur
|title=Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen
|journal=[[Journal für die reine und angewandte Mathematik]]
|volume=139
|year=1911
|pages=155–250
|jfm=42.0154.02
|url=http://www.digizeitschriften.de/resolveppn/GDZPPN002167298
|doi=10.1515/crll.1911.139.155
}}
** {{Citation | last1=Schur | first1=J. | author1-link=Issai Schur | title=On the representation of the symmetric and alternating groups by fractional linear substitutions| doi=10.1023/A:1003772419522 | mr=1820589 | zbl = 0969.20002 | year=2001 | journal=International Journal of Theoretical Physics | issn=0020-7748 | volume=40 | issue=1 | pages=413–458 | postscript= (translation of {{harv|Schur|1911}} by Marc-Felix Otto)}}
* {{citation
 |first=Robert 
 |last=Wilson 
 |author-link=Robert Arnott Wilson 
 |title=The Finite Simple Groups 
 |date=October 31, 2006 
 |url=http://www.maths.qmul.ac.uk/~raw/fsgs.html 
 |chapter=Chapter 2: Alternating groups 
 |chapter-url=http://www.maths.qmul.ac.uk/~raw/fsgs_files/alt.ps 
 |postscript=, 2.7: Covering groups 
 |url-status=dead 
 |archive-url=https://web.archive.org/web/20110522121819/http://www.maths.qmul.ac.uk/~raw/fsgs.html 
 |archive-date=May 22, 2011 
}}
{{refend}}

[[Category:Finite groups]]
[[Category:Permutation groups]]