View source for McLaughlin sporadic group

{{Short description|Sporadic simple group}}
{{Group theory sidebar |Finite}}

In the area of modern algebra known as [[group theory]], the '''McLaughlin group''' McL is a [[sporadic simple group]] of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;898,128,000 = 2<sup>7</sup> &sdot; 3<sup>6</sup> &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 
: ≈ 9{{e|8}}.

==History and properties==
McL is one of the 26 sporadic groups and was discovered by {{harvs|txt|last=McLaughlin|first=Jack|year=1969}} as an index 2 subgroup of a rank 3 permutation group acting on the [[McLaughlin graph]] with {{nowrap|1=275 = 1 + 112 + 162}} vertices.  It fixes a [[Conway group#Other sporadic groups|2-2-3 triangle]] in the [[Leech lattice]] and thus is a subgroup of the [[Conway group]]s <math>\mathrm{Co}_0</math>, <math>\mathrm{Co}_2</math>, and <math>\mathrm{Co}_3</math>. Its [[Schur multiplier]] has order&nbsp;3, and its [[outer automorphism group]] has order&nbsp;2. The group 3.McL:2 is a maximal subgroup of the [[Lyons group]].

McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type 2.A<sub>8</sub>. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A<sub>8</sub>.

==Representations==
In the [[Conway group]] Co<sub>3</sub>, McL has the normalizer McL:2, which is maximal in Co<sub>3</sub>.

McL has 2 classes of maximal subgroups isomorphic to the [[Mathieu group]] M<sub>22</sub>. An outer automorphism interchanges the two classes of M<sub>22</sub> groups. This outer automorphism is realized on McL embedded as a subgroup of Co<sub>3</sub>.

A convenient representation of M<sub>22</sub> is in permutation matrices on the last 22 coordinates; it fixes a 2-2-3 triangle with vertices the origin and the [[Conway group|type 2]] points {{nowrap|1='''x''' = (−3, 1<sup>23</sup>)}} and {{nowrap|1='''y''' = (−4,-4,0<sup>22</sup>)}}'. The triangle's edge {{nowrap|1='''x'''-'''y''' = (1, 5, 1<sup>22</sup>)}} is [[Conway group|type 3]]; it is fixed by a Co<sub>3</sub>. This M<sub>22</sub> is the [[Conway group#Monomial subgroup N of Co0|monomial]], ''and a maximal'', subgroup of a representation of McL.

{{harvtxt|Wilson|2009}} (p.&nbsp;207) shows that the subgroup McL is well-defined. In the [[Leech lattice]], suppose a type 3 point '''v''' is fixed by an instance of <math>\mathrm{Co}_3</math>. Count the type 2 points '''w''' such that the inner product '''v'''·'''w''' = 3 (and thus '''v'''-'''w''' is type 2). He shows their number is {{nowrap|1=552 = 2<sup>3</sup>&sdot;3&sdot;23}} and that this Co<sub>3</sub> is transitive on these '''w'''.

|McL| = |Co3|/552 = 898,128,000.

McL is the only sporadic group to admit irreducible representations of [[Quaternionic representation|quaternionic type]]. It has 2 such representations, one of dimension 3520 and one of dimension 4752.

==Maximal subgroups==
{{harvtxt|Finkelstein|1973}} found the 12 conjugacy classes of maximal subgroups of McL as follows:

{| class="wikitable"
|+ Maximal subgroups of ''McL''
|-
! No. !! Structure !! Order !! Index !! Comments
|-
|  1 || U<sub>4</sub>(3)                              ||align=right|3,265,920<br />=&nbsp;2<sup>7</sup>·3<sup>6</sup>·5·7            ||align=right|    275<br />=&nbsp;5<sup>2</sup>·11                           ||[[point stabilizer]] of its action on the McLaughlin graph
|-
|2,3 || [[Mathieu group M22|M<sub>22</sub>]]          ||align=right|  443,520<br />=&nbsp;2<sup>7</sup>·3<sup>2</sup>·5·7·11         ||align=right|  2,025<br />=&nbsp;3<sup>4</sup>·5<sup>2</sup>                ||two classes, fused by an outer automorphism
|-
|  4 || U<sub>3</sub>(5)                              ||align=right|  126,000<br />=&nbsp;2<sup>4</sup>·3<sup>2</sup>·5<sup>3</sup>·7||align=right|  7,128<br />=&nbsp;2<sup>3</sup>·3<sup>4</sup>·11             ||
|-
|  5 || 3<sup>1+4</sup>:2.S<sub>5</sub>               ||align=right|   58,320<br />=&nbsp;2<sup>4</sup>·3<sup>6</sup>·5              ||align=right| 15,400<br />=&nbsp;2<sup>3</sup>·5<sup>2</sup>·7·11 		  ||normalizer of a subgroup of order 3 (class 3A)
|-
|  6 || 3<sup>4</sup>:[[Mathieu group|M<sub>10</sub>]]||align=right|   58,320<br />=&nbsp;2<sup>4</sup>·3<sup>6</sup>·5              ||align=right| 15,400<br />=&nbsp;2<sup>3</sup>·5<sup>2</sup>·7·11 		  ||
|-
|  7 || L<sub>3</sub>(4):2<sub>2</sub>                ||align=right|   40,320<br />=&nbsp;2<sup>7</sup>·3<sup>2</sup>·5·7            ||align=right| 22,275<br />=&nbsp;3<sup>4</sup>·5<sup>2</sup>·11             ||
|-
|  8 || 2.A<sub>8</sub>                               ||align=right|   40,320<br />=&nbsp;2<sup>7</sup>·3<sup>2</sup>·5·7            ||align=right| 22,275<br />=&nbsp;3<sup>4</sup>·5<sup>2</sup>·11             ||centralizer of involution
|-
|9,10|| 2<sup>4</sup>:A<sub>7</sub>                   ||align=right|   40,320<br />=&nbsp;2<sup>7</sup>·3<sup>2</sup>·5·7            ||align=right| 22,275<br />=&nbsp;3<sup>4</sup>·5<sup>2</sup>·11             ||two classes, fused by an outer automorphism
|-
| 11 || [[Mathieu group M11|M<sub>11</sub>]]          ||align=right|    7,920<br />=&nbsp;2<sup>4</sup>·3<sup>2</sup>·5·11           ||align=right|113,400<br />=&nbsp;2<sup>3</sup>·3<sup>4</sup>·5<sup>2</sup>·7||the subgroup fixed by an outer involution
|-
| 12 || 5{{su|a=l|b=+|p=1+2}}:3:8               ||align=right|    3,000<br />=&nbsp;2<sup>3</sup>·3·5<sup>3</sup>              ||align=right|299,376<br />=&nbsp;2<sup>4</sup>·3<sup>5</sup>·7·11           ||normalizer of a subgroup of order 5 (class 5A)
|}

==Conjugacy classes==
Traces of matrices in a standard 24-dimensional representation of McL are shown. <ref>Conway et al. (1985)</ref> The names of conjugacy classes are taken from the Atlas of Finite Group Representations.<ref>{{Cite web|url=http://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/McLG1-p275B0|title=ATLAS: MCL — Permutation representation on 275 points}}</ref>

Cycle structures in the rank 3 permutation representation, degree 275, of McL are shown.<ref>{{Cite web|url=http://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/McLG1-p275B0|title=ATLAS: MCL — Permutation representation on 275 points}}</ref>

{| class="wikitable" style="margin: 1em auto;"
|-
!Class!!Centraliser order!!No. elements!!Trace!!Cycle type||
|-
| 1A ||	898,128,000 ||1 || 24
|-
| 2A ||40,320 || 3<sup>4</sup> &sdot; 5<sup>2</sup> &sdot; 11 ||8 || 1<sup>35</sup>, 2<sup>120</sup>
|-
| 3A || 29,160 || 2<sup>4</sup> &sdot; 5<sup>2</sup> &sdot; 7 &sdot; 11 || -3 || 1<sup>5</sup>, 3<sup>90</sup>
|-
| 3B ||	972 || 2<sup>3</sup> &sdot; 3 &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 || 6 || 1<sup>14</sup>, 3<sup>87</sup>
|-
| 4A ||	96 || 2<sup>2</sup> &sdot; 3<sup>5</sup> &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 || 4 || 1<sup>7</sup>, 2<sup>14</sup>, 4<sup>60</sup>
|-
| 5A ||	750 || 2<sup>6</sup> &sdot; 3<sup>5</sup> &sdot; &sdot; 7 &sdot; 11 || -1 || 5<sup>55</sup>
|-
| 5B ||	25 || 2<sup>7</sup> &sdot; 3<sup>6</sup> &sdot; 5 &sdot; 7 &sdot; 11 || 4 || 1<sup>5</sup>, 5<sup>54</sup>
|-
| 6A ||	360 || 2<sup>4</sup> &sdot; 3<sup>4</sup> &sdot; 5<sup>2</sup> &sdot; 7 &sdot; 11 || 5 || 1<sup>5</sup>, 3<sup>10</sup>, 6<sup>40</sup>
|-
| 6B ||	36 || 2<sup>5</sup> &sdot; 3<sup>4</sup> &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 || 2 || 1<sup>2</sup>, 2<sup>6</sup>, 3<sup>11</sup>, 6<sup>38</sup>
|-
| 7A ||	14 || 2<sup>6</sup> &sdot; 3<sup>6</sup> &sdot; 5<sup>3</sup> &sdot; 11 || 3 || 1<sup>2</sup>, 7<sup>39</sup> || rowspan = "2"| power equivalent
|-
| 7B ||	14 || 2<sup>6</sup> &sdot; 3<sup>6</sup> &sdot; 5<sup>3</sup> &sdot; 11 || 3 || 1<sup>2</sup>, 7<sup>39</sup>
|-
| 8A ||	8 || 2<sup>4</sup> &sdot; 3<sup>6</sup> &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 || 2 || 1, 2<sup>3</sup>, 4<sup>7</sup>, 8<sup>30</sup>
|-
| 9A ||	27 || 2<sup>7</sup> &sdot; 3<sup>3</sup> &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 || 3 || 1<sup>2</sup>, 3, 9<sup>30</sup> || rowspan = "2"| power equivalent
|-
| 9B ||	27 || 2<sup>7</sup> &sdot; 3<sup>3</sup> &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 || 3 ||  1<sup>2</sup>, 3, 9<sup>30</sup>
|-
| 10A || 10 || 2<sup>6</sup> &sdot; 3<sup>5</sup> &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 || 3 || 5<sup>7</sup>, 10<sup>24</sup>
|-
| 11A || 11 || 2<sup>7</sup> &sdot; 3<sup>6</sup> &sdot; 5<sup>3</sup> &sdot; 7 || 2 || 11<sup>25</sup> || rowspan = "2"| power equivalent
|-
| 11B || 11 || 2<sup>7</sup> &sdot; 3<sup>6</sup> &sdot; 5<sup>3</sup> &sdot; 7 || 2 || 11<sup>25</sup>
|-
| 12A || 12 || 2<sup>5</sup> &sdot; 3<sup>5</sup> &sdot; 5<sup>3</sup> &sdot; 7 &sdot; 11 || 1 || 1, 2<sup>2</sup>, 3<sup>2</sup>, 6<sup>4</sup>, 12<sup>20</sup>
|-
| 14A || 14 || 2<sup>6</sup> &sdot; 3<sup>6</sup> &sdot; 5<sup>3</sup> &sdot; 11 || 1 || 2, 7<sup>5</sup>, 14<sup>17</sup> || rowspan = "2"| power equivalent
|-
| 14B || 14 || 2<sup>6</sup> &sdot; 3<sup>6</sup> &sdot; 5<sup>3</sup> &sdot; 11 || 1 || 2, 7<sup>5</sup>, 14<sup>17</sup>
|-
| 15A || 30 || 2<sup>6</sup> &sdot; 3<sup>5</sup> &sdot; 5<sup>2</sup> &sdot; 7 &sdot; 11 || 2 || 5, 15<sup>18</sup> || rowspan = "2"| power equivalent
|-
| 15B || 30 || 2<sup>6</sup> &sdot; 3<sup>5</sup> &sdot; 5<sup>2</sup> &sdot; 7 &sdot; 11 || 2 || 5, 15<sup>18</sup>
|-
| 30A || 30 || 2<sup>6</sup> &sdot; 3<sup>5</sup> &sdot; 5<sup>2</sup> &sdot; 7 &sdot; 11 || 0 || 5, 15<sup>2</sup>, 30<sup>8</sup> || rowspan = "2"| power equivalent
|-
| 30B || 30 || 2<sup>6</sup> &sdot; 3<sup>5</sup> &sdot; 5<sup>2</sup> &sdot; 7 &sdot; 11 || 0 || 5, 15<sup>2</sup>, 30<sup>8</sup>
|}

==Generalized Monstrous Moonshine==
{{Main|Generalized moonshine}}
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For the [[Conway group#Generalized Monstrous Moonshine|Conway groups]], the relevant McKay–Thompson series is <math>T_{2A}(\tau)</math> and <math>T_{4A}(\tau)</math>.

== References==
{{Reflist}}
* [[John Horton Conway|Conway, J. H]].; Curtis, R. T.; [[Simon P. Norton|Norton, S. P.]]; Parker, R. A.; and [[Robert Arnott Wilson|Wilson, R. A.]]: "''Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups.''"  Oxford, England 1985.
*{{Citation | last1=Finkelstein | first1=Larry | title=The maximal subgroups of Conway's group C<sub>3</sub> and McLaughlin's group | doi=10.1016/0021-8693(73)90075-6 |mr=0346046 | year=1973 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=25 | pages=58–89| doi-access=free }}
* {{Citation | last1=Griess | first1=Robert L. Jr. | author1-link=R. L. Griess | title=Twelve sporadic groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-62778-4 |mr=1707296 | year=1998 | doi=10.1007/978-3-662-03516-0}}
*{{Citation | last1=McLaughlin | first1=Jack  | editor1-last=Brauer | editor1-first=R. | editor1-link=Richard Brauer | editor2-last=Sah | editor2-first=Chih-han | title=Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968) | publisher=Benjamin, New York | year=1969 | chapter=A simple group of order 898,128,000 | pages=109–111 | mr=0242941}}
*{{Citation | last1=Wilson | first1=Robert A. | authorlink = Robert Arnott Wilson | title=The finite simple groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] 251 | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | year=2009 | zbl=1203.20012 | volume=251  }}

==External links==
* [http://mathworld.wolfram.com/McLaughlinGroup.html MathWorld: McLaughlin group]
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/McL/  Atlas of Finite Group Representations: McLaughlin group]

{{DEFAULTSORT:Mclaughlin Group (Mathematics)}}
[[Category:Sporadic groups]]