Editing Lyons group

{{Short description|Sporadic simple group}}
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In the area of modern algebra known as [[group theory]], the '''Lyons group''' ''Ly'' or '''Lyons-Sims group''' ''LyS'' is a [[sporadic simple group]] of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;51,765,179,004,000,000
: = 2<sup>8</sup>{{·}}3<sup>7</sup>{{·}}5<sup>6</sup>{{·}}7{{·}}11{{·}}31{{·}}37{{·}}67
: ≈ 5{{e|16}}.

==History==
''Ly'' is one of the 26 sporadic groups and was discovered by [[Richard Lyons (mathematician)|Richard Lyons]] and [[Charles Sims (mathematician)|Charles Sims]] in 1972-73.   Lyons characterized 51765179004000000 as the unique possible order of any finite simple group where the [[centralizer]] of some [[Involution (mathematics)|involution]] is [[isomorphic]] to the nontrivial central extension of the [[alternating group]] A<sub>11</sub> of degree 11 by the [[cyclic group]] C<sub>2</sub>. {{harvtxt|Sims|1973}} proved the existence of such a group and its uniqueness up to isomorphism with a combination of permutation group theory and machine calculations. 

When the [[McLaughlin group (mathematics)|McLaughlin sporadic group]] was discovered, it was noticed that a centralizer of one of its involutions was the perfect [[Double covering group|double cover]] of the [[alternating group]] ''A''<sub>8</sub>. This suggested considering  the double covers of the other alternating groups ''A''<sub>''n''</sub> as possible centralizers of involutions in simple groups. The cases ''n'' ≤ 7 are ruled out by the [[Brauer–Suzuki theorem]], the case ''n'' = 8 leads to the McLaughlin group, the case ''n'' = 9 was ruled out by [[Zvonimir Janko]], Lyons himself ruled out the case ''n'' = 10 and found the Lyons group for ''n'' = 11, while the cases ''n'' ≥ 12 were ruled out by [[John Griggs Thompson|J.G. Thompson]] and [[Ronald Solomon]].

The [[Schur multiplier]] and the [[outer automorphism group]] are both [[Trivial group|trivial]]. 

Since 37 and 67 are not [[supersingular prime (moonshine theory)|supersingular]] primes, the Lyons group cannot be a [[subquotient]] of the [[monster group]]. Thus it is one of the 6 sporadic groups called the [[pariah group|pariahs]].

==Representations==

{{harvtxt|Meyer|Neutsch|Parker|1985}} showed that the Lyons group has a [[modular representation]] of dimension 111 over the field of five elements, which is the smallest dimension of any faithful linear representation and is one of the easiest ways of calculating with it. It has also been given by several complicated presentations in terms of generators and relations, for instance those given by {{harvtxt|Sims|1973}} or {{harvtxt|Gebhardt|2000}}.

The smallest faithful [[permutation representation]] is a rank 5 permutation representation on 8835156 points with stabilizer G<sub>2</sub>(5). There is also a slightly larger rank 5 permutation representation on 9606125 points with stabilizer 3.McL:2.

The degrees of [[irreducible representation|irreducible representations]] of the Lyons group are 1, 2480, 2480, 45694, 48174,&nbsp;... {{OEIS|id=A003917}}.

==Maximal subgroups==
{{harvtxt|Wilson|1985}} found the 9 conjugacy classes of maximal subgroups of ''Ly'' as follows:
{| class="wikitable"
|+ Maximal subgroups of ''Ly''
|-
! No. !! Structure !! Order !! Index !! Comments
|-
|1||G<sub>2</sub>(5)                              ||align=right|5,859,000,000<br />=&nbsp;2<sup>6</sup>·3<sup>3</sup>·5<sup>6</sup>·7·31||align=right|         8,835,156<br />=&nbsp;2<sup>2</sup>·3<sup>4</sup>·11·37·67                || 
|-                                                                                                                                                                                                                                       
|2||3<sup>·</sup>McL:2                            ||align=right|5,388,768,000<br />=&nbsp;2<sup>8</sup>·3<sup>7</sup>·5<sup>3</sup>·7·11||align=right|         9,606,125<br />=&nbsp;5<sup>3</sup>·31·37·67                              || normalizer of a subgroup of order 3 (class 3A)
|-                                                                                                                                                                                                                                       
|3||5<sup>3·</sup>L<sub>3</sub>(5)                ||align=right|   46,500,000<br />=&nbsp;2<sup>5</sup>·3·5<sup>6</sup>·31              ||align=right|     1,113,229,656<br />=&nbsp;2<sup>3</sup>·3<sup>6</sup>·7·11·37·67              ||
|-                                                                                                                                                                                                                                       
|4||2<sup>·</sup>A<sub>11</sub>                   ||align=right|   39,916,800<br />=&nbsp;2<sup>8</sup>·3<sup>4</sup>·5<sup>2</sup>·7·11||align=right|     1,296,826,875<br />=&nbsp;3<sup>3</sup>·5<sup>4</sup>·31·37·67                || centralizer of an involution
|-                                                                                                                                                                                                                                       
|5||5<sup>1+4</sup>:4.S<sub>6</sub>               ||align=right|    9,000,000<br />=&nbsp;2<sup>6</sup>·3<sup>2</sup>·5<sup>6</sup>     ||align=right|     5,751,686,556<br />=&nbsp;2<sup>2</sup>·3<sup>5</sup>·7·11·31·37·67           || normalizer of a subgroup of order 5 (class 5A)
|-                                                                                                                                                                                                                                       
|6||3<sup>5</sup>:(2 × M<sub>11</sub>)            ||align=right|    3,849,120<br />=&nbsp;2<sup>5</sup>·3<sup>7</sup>·5·11              ||align=right|    13,448,575,000<br />=&nbsp;2<sup>3</sup>·5<sup>5</sup>·7·31·37·67              ||
|-                                                                                                                                                                                                                                       
|7||3<sup>2+4</sup>:2.A<sub>5</sub>.D<sub>8</sub> ||align=right|      699,840<br />=&nbsp;2<sup>6</sup>·3<sup>7</sup>·5                 ||align=right|    73,967,162,500<br />=&nbsp;2<sup>2</sup>·5<sup>5</sup>·7·11·31·37·67           ||
|-                                                                                                                                                                                                                                       
|8||67:22                                         ||align=right|        1,474<br />=&nbsp;2·11·67                                       ||align=right|35,118,846,000,000<br />=&nbsp;2<sup>7</sup>·3<sup>7</sup>·5<sup>6</sup>·7·31·37   ||
|-                                                                                                                                                                                                                                       
|9||37:18                                         ||align=right|          666<br />=&nbsp;2·3<sup>2</sup>·37                            ||align=right|77,725,494,000,000<br />=&nbsp;2<sup>7</sup>·3<sup>5</sup>·5<sup>6</sup>·7·11·31·67||
|}

== References ==

* [[Richard Lyons (mathematician)|Richard Lyons]] (1972,5) "Evidence for a new finite simple group", [[Journal of Algebra]] 20:540&ndash;569 and 34:188&ndash;189.
* {{cite journal | last1 = Gebhardt | first1 = Volker | year = 2000 | title = Two short presentations for Lyons' sporadic simple group | journal = Experimental Mathematics | volume = 9 | issue = 3| pages = 333–8 | doi=10.1080/10586458.2000.10504410| s2cid = 8361971 | url = http://projecteuclid.org/euclid.em/1045604668 }}
* {{Citation | last3=Parker | first3=Richard | last2=Neutsch | first2=Wolfram | last1=Meyer | first1=Werner | title=The minimal 5-representation of Lyons' sporadic group | doi=10.1007/BF01455926 |mr=794089 | year=1985 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=272 | issue=1 | pages=29–39| s2cid=120696430 }}
* {{Citation | last1=Sims | first1=Charles C. | author-link=Charles Sims (mathematician) |title=Finite groups '72 (Proc. Gainesville Conf., Univ. Florida, Gainesville, Fla., 1972) | publisher=North-Holland | location=Amsterdam | series= North-Holland Math. Studies |mr=0354881 | year=1973 | volume=7 | chapter=The existence and uniqueness of Lyons' group | pages=138–141}}
* {{Citation | last1=Wilson | first1=Robert A. | title=The maximal subgroups of the Lyons group | doi= 10.1017/S0305004100063003  |mr=778677 | year=1985 | journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] | issn=0305-0041 | volume=97 | issue=3 | pages=433–436| s2cid=119577612 }}

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

[[Category:Sporadic groups]]