View source for Rudvalis group

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

In the area of modern algebra known as [[group theory]], the '''Rudvalis group''' ''Ru'' is a [[sporadic simple group]] of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;145,926,144,000 = 2<sup>14</sup>{{·}}3<sup>3</sup>{{·}}5<sup>3</sup>{{·}}7{{·}}13{{·}}29
: ≈ 1{{e|11}}.

==History==
''Ru'' is one of the 26 sporadic groups and was found by {{Harvard citations |last=Rudvalis |first=Arunas |author-link=Arunas Rudvalis |year=1973 |year2=1984 |txt=yes}} and constructed by {{Harvard citations |last1 = Conway|first1 = John H.|author1-link = John H. Conway|last2 = Wales|first2 = David B.|year=1973 |txt=yes}}. Its [[Schur multiplier]] has order 2, and its [[outer automorphism group]] is trivial.

In 1982 [[Robert Griess]] showed that ''Ru'' cannot be a [[subquotient]] of the [[monster group]].<ref>Griess (1982)</ref>  Thus it is one of the 6 sporadic groups called the [[Pariah group|pariah]]s.

== Properties ==

The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer being the [[Ree group]] 
<sup>2</sup>''F''<sub>4</sub>(2), the automorphism group of the [[Tits group]]. This representation implies a [[strongly regular graph]] srg(4060, 2304, 1328, 1280). That is, each vertex has 2304 neighbors and 1755 non-neighbors, any two adjacent vertices have 1328 common neighbors, while any two non-adjacent ones have 1280 {{Harvard citations|last=Griess|year=1998|loc=p. 125}}.

Its [[Double covering group|double cover]] acts on a 28-dimensional lattice over the [[Gaussian integer]]s. The lattice has 4×4060 minimal vectors; if minimal vectors are identified whenever one is 1, ''i'', –1, or –''i'' times another, then the 4060 equivalence classes can be identified with the points of the rank 3 permutation representation. Reducing this lattice modulo the [[principal ideal]]

:<math>(1 + i)\ </math>

gives an action of the Rudvalis group on a 28-dimensional vector space over the field <math>\mathbb F_2</math> with 2 elements.  Duncan (2006) used the 28-dimensional lattice to construct a [[vertex operator algebra]] acted on by the double cover.

Alternatively, the double cover can be defined abstractly, by starting with the graph and lifting Ru to 2Ru in the double cover 2A<sub>4060</sub>. This is because 1 of the conjugacy classes of [[Involution (mathematics)#Group theory|involution]]s does not fix any points. Such an involution partitions the 4060 points of the graph into 2030 pairs, which can be regarded as 1015 double transpositions in the [[alternating group]] A<sub>4060</sub>. Since 1015 is odd, these involutions are lifted to order 4 elements in the double cover 2A<sub>4060</sub>. For more information, see [[Covering groups of the alternating and symmetric groups]].

{{harvtxt|Parrott|1976}} characterized the Rudvalis group by the centralizer of a central involution. {{harvtxt|Aschbacher|Smith|2004}} gave another characterization as part of their identification of the Rudvalis group as one of the [[quasithin group]]s.

== Maximal subgroups ==
{{harvtxt|Wilson|1984}} found the 15 conjugacy classes of maximal subgroups of ''Ru'' as follows:

{| class="wikitable"
|+ Maximal subgroups of ''Ru''
|-
! No. !! Structure !! Order !! Index !! Comments
|-
| 1||<sup>2</sup>F<sub>4</sub>(2) = <sup>2</sup>F<sub>4</sub>(2)'.2||align=right|35,942,400<br />=&nbsp;2<sup>12</sup>·3<sup>3</sup>·5<sup>2</sup>·13||align=right|      4,060<br />=&nbsp;2<sup>2</sup>·5·7·29                          ||
|-
| 2||2<sup>6</sup>.U<sub>3</sub>(3).2                              ||align=right|   774,144<br />=&nbsp;2<sup>12</sup>·3<sup>3</sup>·7               ||align=right|    188,500<br />=&nbsp;2<sup>2</sup>·5<sup>3</sup>·13·29             ||
|-
| 3||(2<sup>2</sup> × Sz(8)):3                                     ||align=right|   349,440<br />=&nbsp;2<sup>8</sup>·3·5·7·13                       ||align=right|    417,600<br />=&nbsp;2<sup>6</sup>·3<sup>2</sup>·5<sup>2</sup>·29  ||
|-
| 4||2<sup>3+8</sup>:L<sub>3</sub>(2)                              ||align=right|   344,064<br />=&nbsp;2<sup>14</sup>·3·7                           ||align=right|    424,125<br />=&nbsp;3<sup>2</sup>·5<sup>3</sup>·13·29             ||
|-
| 5||U<sub>3</sub>(5):2                                            ||align=right|   252,000<br />=&nbsp;2<sup>5</sup>·3<sup>2</sup>·5<sup>3</sup>·7  ||align=right|    579,072<br />=&nbsp;2<sup>9</sup>·3·13·29                         ||
|-
| 6||2<sup>1+4+6</sup>.S<sub>5</sub>                               ||align=right|   245,760<br />=&nbsp;2<sup>14</sup>·3·5                           ||align=right|    593,775<br />=&nbsp;3<sup>2</sup>·5<sup>2</sup>·7·13·29           || centralizer of an involution of class 2A
|-
| 7||L<sub>2</sub>(25).2<sup>2</sup>                             ||align=right|    31,200<br />=&nbsp;2<sup>5</sup>·3·5<sup>2</sup>·13             ||align=right|  4,677,120<br />=&nbsp;2<sup>9</sup>·3<sup>2</sup>·5·7·29            ||
|-
| 8||A<sub>8</sub>                                                 ||align=right|    20,160<br />=&nbsp;2<sup>6</sup>·3<sup>2</sup>·5·7              ||align=right|  7,238,400<br />=&nbsp;2<sup>8</sup>·3·5<sup>2</sup>·13·29           ||
|-
| 9||L<sub>2</sub>(29)                                           ||align=right|    12,180<br />=&nbsp;2<sup>2</sup>·3·5·7·29                       ||align=right| 11,980,800<br />=&nbsp;2<sup>12</sup>·3<sup>2</sup>·5<sup>2</sup>·13 ||
|-
|10||5<sup>2</sup>:4.S<sub>5</sub>                                 ||align=right|    12,000<br />=&nbsp;2<sup>5</sup>·3·5<sup>3</sup>                ||align=right| 12,160,512<br />=&nbsp;2<sup>9</sup>·3<sup>2</sup>·7·13·29           ||
|-
|11||3<sup>&thinsp;·</sup>A<sub>6</sub>.2<sup>2</sup>                                 ||align=right|     4,320<br />=&nbsp;2<sup>5</sup>·3<sup>3</sup>·5                ||align=right| 33,779,200<br />=&nbsp;2<sup>9</sup>·5<sup>2</sup>·7·13·29           || normalizer of a subgroup of order 3
|-
|12||5{{su|a=l|b=+|p=1+2}}:&thinsp;[2<sup>5</sup>]                               ||align=right|     4,000<br />=&nbsp;2<sup>5</sup>·5<sup>3</sup>                  ||align=right| 36,481,536<br />=&nbsp;2<sup>9</sup>·3<sup>3</sup>·7·13·29           || normalizer of a subgroup of order 5 (class 5A)
|-
|13||L<sub>2</sub>(13):2                                           ||align=right|     2,184<br />=&nbsp;2<sup>3</sup>·3·7·13                         ||align=right| 66,816,000<br />=&nbsp;2<sup>11</sup>·3<sup>2</sup>·5<sup>3</sup>·29 ||
|-
|14||A<sub>6</sub>.2<sup>2</sup>                                   ||align=right|     1,440<br />=&nbsp;2<sup>5</sup>·3<sup>2</sup>·5                ||align=right|101,337,600<br />=&nbsp;2<sup>9</sup>·3·5<sup>2</sup>·7·13·29         ||
|-
|15||5:4 × A<sub>5</sub>                                           ||align=right|     1,200<br />=&nbsp;2<sup>4</sup>·3·5<sup>2</sup>                ||align=right|121,605,120<br />=&nbsp;2<sup>10</sup>·3<sup>2</sup>·5·7·13·29        || normalizer of a subgroup of order 5 (class 5B)
|-
|}

==References ==
{{Reflist}}

*{{Citation | last1=Aschbacher | first1=Michael | author1-link=Michael Aschbacher | last2=Smith | first2=Stephen D. | title=The classification of quasithin groups. I Structure of Strongly Quasithin K-groups | url=http://www.ams.org/bookstore-getitem/item=SURV-111 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-3410-7 | mr=2097623 | year=2004 | volume=111}}
* {{Citation
|last1 = Conway
|first1 = John H.
|author1-link = John H. Conway
|last2 = Wales
|first2 = David B.
|title = The construction of the Rudvalis simple group of order 145926144000
|journal = [[Journal of Algebra]]
|issue = 3
|year = 1973
|pages = 538–548
|doi = 10.1016/0021-8693(73)90063-X
|volume = 27
|doi-access = free
}}
* {{cite arXiv
|author = John F. Duncan
|title = Moonshine for Rudvalis's sporadic group
|year = 2008
|eprint = math/0609449v1
}}
* {{Citation
|last = Griess
|first = Robert L.
|author-link = Robert Griess
|title = The Friendly Giant
|journal = Inventiones Mathematicae
|issue = 1
|year = 1982
|pages = 1–102
|doi = 10.1007/BF01389186
|volume = 69
|bibcode = 1982InMat..69....1G
|url = https://deepblue.lib.umich.edu/bitstream/2027.42/46608/1/222_2005_Article_BF01389186.pdf
|hdl = 2027.42/46608
|hdl-access = free
}}
* {{Citation
|last = Griess
|first = Robert L.
|title = Twelve Sporadic Groups
|year = 1998
|publisher = Springer-Verlag
}}
*{{Citation | last1=Parrott | first1=David | title=A characterization of the Rudvalis simple group | doi=10.1112/plms/s3-32.1.25  | mr=0390043 | year=1976 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=32 | issue=1 | pages=25–51}}
* {{Citation
|last = Rudvalis
|first = Arunas
|author-link = Arunas Rudvalis
|title = A new simple group of order 2<sup>14</sup> 3<sup>3</sup> 5<sup>3</sup> 7 13 29
|journal = [[Notices of the American Mathematical Society]]
|issue = 20
|year = 1973
|pages = A–95
}}
*{{Citation | last1=Rudvalis | first1=Arunas | title=A rank 3 simple group of order 2¹⁴3³5³7.13.29. I | doi=10.1016/0021-8693(84)90063-2 |mr=727376 | year=1984 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=86 | issue=1 | pages=181–218| doi-access= }}
*{{Citation | last1=Rudvalis | first1=Arunas | title=A rank 3 simple group G of order 2¹⁴3³5³7.13.29. II. Characters of G and Ĝ | doi=10.1016/0021-8693(84)90064-4 |mr=727377 | year=1984 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=86 | issue=1 | pages=219–258| doi-access=free }}
*{{Citation | last1=Wilson | first1=Robert A. | title=The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits | doi=10.1112/plms/s3-48.3.533 | mr=735227 | year=1984 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=48 | issue=3 | pages=533–563}}

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

[[Category:Sporadic groups]]