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{{Group theory sidebar |Finite}} In the area of modern algebra known as [[group theory]], the '''Conway groups''' are the three [[sporadic simple group]]s [[Conway group Co1|Co<sub>1</sub>]], [[Conway group Co2|Co<sub>2</sub>]] and [[Conway group Co3|Co<sub>3</sub>]] along with the related finite group [[Leech lattice#Symmetries|Co<sub>0</sub>]] introduced by {{harvs |authorlink=John Horton Conway |last=Conway |year1=1968 |year2=1969}}. The largest of the Conway groups, '''Co<sub>0</sub>''', is the [[Automorphism group|group of automorphisms]] of the [[Leech lattice]] Λ with respect to addition and [[inner product]]. It has [[Order (group theory)|order]] : {{val|fmt=commas|8,315,553,613,086,720,000}} but it is not a simple group. The simple group '''[[Conway group Co1|Co<sub>1</sub>]]''' of order : {{val|fmt=commas|4,157,776,806,543,360,000}} = 2<sup>21</sup>{{·}}3<sup>9</sup>{{·}}5<sup>4</sup>{{·}}7<sup>2</sup>{{·}}11{{·}}13{{·}}23 is defined as the quotient of '''Co<sub>0</sub>''' by its [[center of a group|center]], which consists of the scalar matrices ±1. The groups '''[[Conway group Co2|Co<sub>2</sub>]]''' of order :{{val|fmt=commas|42,305,421,312,000}} = 2<sup>18</sup>{{·}}3<sup>6</sup>{{·}}5<sup>3</sup>{{·}}7{{·}}11{{·}}23 and '''[[Conway group Co3|Co<sub>3</sub>]]''' of order :{{val|fmt=commas|495,766,656,000}} = 2<sup>10</sup>{{·}}3<sup>7</sup>{{·}}5<sup>3</sup>{{·}}7{{·}}11{{·}}23 consist of the automorphisms of Λ fixing a lattice vector of type 2 and type 3, respectively. As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co<sub>1</sub>. The '''inner product''' on the Leech lattice is defined as 1/8 the [[dot product|sum of the products]] of respective co-ordinates of the two multiplicand vectors; it is an integer. The '''square norm''' of a vector is its inner product with itself, always an even integer. It is common to speak of the '''type''' of a Leech lattice vector: half the square norm. Subgroups are often named in reference to the ''types'' of relevant fixed points. This lattice has no vectors of type 1. ==History== {{harvs|txt|first=Thomas |last=Thompson|year=1983}} relates how, in about 1964, [[John Leech (mathematician)|John Leech]] investigated close packings of spheres in Euclidean spaces of large dimension. One of Leech's discoveries was a lattice packing in 24-space, based on what came to be called the Leech lattice Λ. He wondered whether his lattice's symmetry group contained an interesting simple group, but felt he needed the help of someone better acquainted with group theory. He had to do much asking around because the mathematicians were pre-occupied with agendas of their own. [[John Horton Conway|John Conway]] agreed to look at the problem. [[John G. Thompson]] said he would be interested if he were given the order of the group. Conway expected to spend months or years on the problem, but found results in just a few sessions. {{harvtxt|Witt|1998|loc= page 329}} stated that he found the Leech lattice in 1940 and hinted that he calculated the order of its automorphism group Co<sub>0</sub>. ==Monomial subgroup N of Co<sub>0</sub>== Conway started his investigation of Co<sub>0</sub> with a subgroup he called '''N''', a [[holomorph (mathematics)|holomorph]] of the (extended) [[binary Golay code#Mathematical definition|binary Golay code]] (as [[diagonal matrices]] with 1 or −1 as diagonal elements) by the [[Mathieu group M24|Mathieu group M<sub>24</sub>]] (as [[permutation matrices]]). {{nowrap|'''N''' ≈ 2<sup>12</sup>:M<sub>24</sub>}}. A standard [[binary Golay code#A convenient representation|representation]], used throughout this article, of the binary Golay code arranges the 24 co-ordinates so that 6 consecutive blocks (tetrads) of 4 constitute a [[Mathieu group M24#Sextet subgroup|sextet]]. The matrices of Co<sub>0</sub> are [[orthogonal#Definitions|orthogonal]]; i. e., they leave the inner product invariant. The [[matrix (mathematics)#Invertible matrix and its inverse|inverse]] is the [[transpose]]. Co<sub>0</sub> has no matrices of [[matrix (mathematics)#Determinant|determinant]] −1. The Leech lattice can easily be defined as the '''Z'''-[[module (mathematics)|module]] generated by the set Λ<sub>2</sub> of all vectors of type 2, consisting of : (4, 4, 0<sup>22</sup>) : (2<sup>8</sup>, 0<sup>16</sup>) : (−3, 1<sup>23</sup>) and their images under '''N'''. Λ<sub>2</sub> under '''N''' falls into 3 [[Group action (mathematics)#Orbits and stabilizers|orbit]]s of sizes [[Leech lattice#Geometry|1104, 97152, and 98304]]. Then {{nowrap|1={{abs|Λ<sub>2</sub>}} = {{val|fmt=commas|196560}} = 2<sup>4</sup>⋅3<sup>3</sup>⋅5⋅7⋅13}}. Conway strongly suspected that Co<sub>0</sub> was [[Group action (mathematics)#Remarkable properties of actions|transitive]] on Λ<sub>2</sub>, and indeed he found a new matrix, not [[generalized permutation matrix|monomial]] and not an integer matrix. Let ''η'' be the 4-by-4 matrix :<math>\frac{1}{2}\begin{pmatrix} 1 & -1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ -1 & -1 & 1 & -1 \\ -1 & -1 & -1 & 1 \end{pmatrix}</math> Now let ζ be a block sum of 6 matrices: odd numbers each of ''η'' and −''η''.<ref>Griess, p. 97.</ref><ref>Thomas Thompson, pp. 148–152.</ref> ''ζ'' is a [[symmetric matrix|symmetric]] and orthogonal matrix, thus an [[involution (mathematics)#group theory|involution]]. Some experimenting shows that it interchanges vectors between different orbits of '''N'''. To compute |Co<sub>0</sub>| it is best to consider Λ<sub>4</sub>, the set of vectors of type 4. Any type 4 vector is one of exactly 48 type 4 vectors congruent to each other modulo 2Λ, falling into 24 orthogonal pairs {{nowrap|{''v'', –''v''}.}} A set of 48 such vectors is called a '''frame''' or '''cross'''. '''N''' has as an [[Group action (mathematics)#Orbits and stabilizers|orbit]] a standard frame of 48 vectors of form (±8, 0<sup>23</sup>). The subgroup fixing a given frame is a [[conjugacy class#Conjugacy of subgroups and general subsets|conjugate]] of '''N'''. The group 2<sup>12</sup>, isomorphic to the Golay code, acts as sign changes on vectors of the frame, while M<sub>24</sub> permutes the 24 pairs of the frame. Co<sub>0</sub> can be shown to be [[Group action (mathematics)#Remarkable properties of actions|transitive]] on Λ<sub>4</sub>. Conway multiplied the order 2<sup>12</sup>|M<sub>24</sub>| of '''N''' by the number of frames, the latter being equal to the quotient {{nowrap|1={{abs|Λ<sub>4</sub>}}/48 = {{val|fmt=commas|8,252,375}} = 3<sup>6</sup>⋅5<sup>3</sup>⋅7⋅13}}. That product is the order of ''any'' subgroup of Co<sub>0</sub> that properly contains '''N'''; hence '''N''' is a maximal subgroup of Co<sub>0</sub> and contains 2-Sylow subgroups of Co<sub>0</sub>. '''N''' also is the subgroup in Co<sub>0</sub> of all matrices with integer components. Since Λ includes vectors of the shape {{nowrap|(±8, 0<sup>23</sup>)}}, Co<sub>0</sub> consists of rational matrices whose denominators are all divisors of 8. The smallest non-trivial representation of Co<sub>0</sub> over any field is the 24-dimensional one coming from the Leech lattice, and this is faithful over fields of characteristic other than 2. ==Involutions in Co<sub>0</sub>== Any [[involution (mathematics)#group theory|involution]] in Co<sub>0</sub> can be shown to be [[Conjugacy class|conjugate]] to an element of the Golay code. Co<sub>0</sub> has 4 conjugacy classes of involutions. A permutation matrix of shape 2<sup>12</sup> can be shown to be conjugate to a [[Binary Golay code#Mathematical definition|dodecad]]. Its centralizer has the form 2<sup>12</sup>:M<sub>12</sub> and has conjugates inside the monomial subgroup. Any matrix in this conjugacy class has trace 0. A permutation matrix of shape 2<sup>8</sup>1<sup>8</sup> can be shown to be conjugate to an [[Binary Golay code#Mathematical definition|octad]]; it has trace 8. This and its negative (trace −8) have a common centralizer of the form {{nowrap|(2<sup>1+8</sup>×2).O<sub>8</sub><sup>+</sup>(2)}}, a subgroup maximal in Co<sub>0</sub>. ==Sublattice groups== Conway and Thompson found that four recently discovered sporadic simple groups, described in conference proceedings {{harv|Brauer|Sah|1969}}, were isomorphic to subgroups or quotients of subgroups of Co<sub>0</sub>. Conway himself employed a notation for stabilizers of points and subspaces where he prefixed a dot. Exceptional were '''.0''' and '''.1''', being Co<sub>0</sub> and Co<sub>1</sub>. For integer {{nowrap|'''n''' ≥ 2}} let '''.n''' denote the stabilizer of a point of type '''n''' (see above) in the Leech lattice. Conway then named stabilizers of planes defined by triangles having the origin as a vertex. Let '''.hkl''' be the pointwise stabilizer of a triangle with edges (differences of vertices) of types '''h''', '''k''' and '''l'''. The triangle is commonly called an '''h-k-l triangle'''. In the simplest cases Co<sub>0</sub> is transitive on the points or triangles in question and stabilizer groups are defined up to conjugacy. Conway identified '''.322''' with the '''[[McLaughlin group (mathematics)|McLaughlin group]]''' McL (order {{val|fmt=commas|898,128,000}}) and '''.332''' with the '''[[Higman–Sims group]]''' HS (order {{val|fmt=commas|44,352,000}}); both of these had recently been discovered. Here is a table<ref>Conway & Sloane (1999), p. 291</ref><ref>Griess (1998), p. 126</ref> of some sublattice groups: {| class="wikitable" style="margin:1em auto;" |- ! Name ! Order ! Structure ! Example vertices |- | •2 || 2<sup>18</sup> 3<sup>6</sup> 5<sup>3</sup> 7 11 23 || Co<sub>2</sub> || (−3, 1<sup>23</sup>) |- | •3 || 2<sup>10</sup> 3<sup>7</sup> 5<sup>3</sup> 7 11 23 || Co<sub>3</sub> || (5, 1<sup>23</sup>) |- | •4 || 2<sup>18</sup> 3<sup>2</sup> 5 7 11 23 || 2<sup>11</sup>:M<sub>23</sub> ||(8, 0<sup>23</sup>) |- | •222 || 2<sup>15</sup> 3<sup>6</sup> 5 7 11 || PSU<sub>6</sub>(2) ≈ [[Fischer group#3-transposition groups|Fi<sub>21</sub>]] || (4, −4, 0<sup>22</sup>), (0, −4, 4, 0<sup>21</sup>) |- | •322 || 2<sup>7</sup> 3<sup>6</sup> 5<sup>3</sup> 7 11 || McL || (5, 1<sup>23</sup>),(4, 4, 0<sup>22</sup>) |- | •332 || 2<sup>9</sup> 3<sup>2</sup> 5<sup>3</sup> 7 11 || HS || (5, 1<sup>23</sup>), (4, −4, 0<sup>22</sup>) |- | •333 || 2<sup>4</sup> 3<sup>7</sup> 5 11 || 3<sup>5</sup> M<sub>11</sub> || (5, 1<sup>23</sup>), (0, 2<sup>12</sup>, 0<sup>11</sup>) |- | •422 || 2<sup>17</sup> 3<sup>2</sup> 5 7 11 || 2<sup>10</sup>:M<sub>22</sub> || (8, 0<sup>23</sup>), (4, 4, 0<sup>22</sup>) |- | •432 || 2<sup>7</sup> 3<sup>2</sup> 5 7 11 23 ||M<sub>23</sub> || (8, 0<sup>23</sup>), (5, 1<sup>23</sup>) |- | •433 || 2<sup>10</sup> 3<sup>2</sup> 5 7 || 2<sup>4</sup>.A<sub>8</sub> || (8, 0<sup>23</sup>), (4, 2<sup>7</sup>, −2, 0<sup>15</sup>) |- | •442 || 2<sup>12</sup> 3<sup>2</sup> 5 7 || 2<sup>1+8</sup>.A<sub>7</sub> || (8, 0<sup>23</sup>), (6, −2<sup>7</sup>, 0<sup>16</sup>) |- | •443 || 2<sup>7</sup> 3<sup>2</sup> 5 7 || M<sub>21</sub>:2 ≈ PSL<sub>3</sub>(4):2 || (8, 0<sup>23</sup>), (5, −3, −3, 1<sup>21</sup>) |} ==Two other sporadic groups== Two sporadic subgroups can be defined as quotients of stabilizers of structures on the Leech lattice. Identifying '''R'''<sup>24</sup> with '''C'''<sup>12</sup> and Λ with : <math>\mathbf{Z}\left[e^{\frac{2}{3}\pi i}\right]^{12},</math> the resulting automorphism group (i.e., the group of Leech lattice automorphisms preserving the [[complex structure on a real vector space|complex structure]]) when divided by the six-element group of complex scalar matrices, gives the '''[[Suzuki sporadic group|Suzuki group]]''' Suz (order {{val|fmt=commas|448,345,497,600}}). This group was discovered by [[Michio Suzuki (mathematician)|Michio Suzuki]] in 1968. A similar construction gives the '''[[Hall–Janko group]]''' J<sub>2</sub> (order {{val|fmt=commas|604,800}}) as the quotient of the group of [[quaternion]]ic automorphisms of Λ by the group ±1 of scalars. The seven simple groups described above comprise what [[Robert Griess]] calls the ''second generation of the Happy Family'', which consists of the 20 sporadic simple groups found within the [[Monster group]]. Several of the seven groups contain at least some of the five [[Mathieu groups]], which comprise the ''first generation''. ==Suzuki chain of product groups== Co<sub>0</sub> has 4 conjugacy classes of elements of order 3. In M<sub>24</sub> an element of shape 3<sup>8</sup> generates a group normal in a copy of S<sub>3</sub>, which commutes with a simple subgroup of order 168. A [[Binary Golay code#A convenient representation|direct product]] {{nowrap|PSL(2,7) × S<sub>3</sub>}} in M<sub>24</sub> permutes the octads of a [[Mathieu group M24#Trio subgroup|trio]] and permutes 14 dodecad diagonal matrices in the monomial subgroup. In Co<sub>0</sub> this monomial normalizer {{nowrap|2<sup>4</sup>:PSL(2,7) × S<sub>3</sub>}} is expanded to a maximal subgroup of the form {{nowrap|2.A<sub>9</sub> × S<sub>3</sub>}}, where 2.A<sub>9</sub> is the double cover of the alternating group A<sub>9</sub>. John Thompson pointed out it would be fruitful to investigate the normalizers of smaller subgroups of the form 2.A<sub>n</sub> {{harv|Conway|1971|loc=p. 242}}. Several other maximal subgroups of Co<sub>0</sub> are found in this way. Moreover, two sporadic groups appear in the resulting chain. There is a subgroup {{nowrap|2.A<sub>8</sub> × S<sub>4</sub>}}, the only one of this chain not maximal in Co<sub>0</sub>. Next there is the subgroup {{nowrap|(2.A<sub>7</sub> × PSL<sub>2</sub>(7)):2}}. Next comes {{nowrap|(2.A<sub>6</sub> × SU<sub>3</sub>(3)):2}}. The unitary group SU<sub>3</sub>(3) (order {{val|fmt=commas|6048}}) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is {{nowrap|(2.A<sub>5</sub> o 2.HJ):2}}, in which the [[Hall–Janko group]] HJ makes its appearance. The aforementioned graph expands to the [[Hall–Janko graph]], with 100 vertices. Next comes {{nowrap|(2.A<sub>4</sub> o 2.G<sub>2</sub>(4)):2}}, G<sub>2</sub>(4) being an exceptional [[G2 (mathematics)|group of Lie type]]. The chain ends with 6.Suz:2 (Suz=[[Suzuki sporadic group]]), which, as mentioned above, respects a complex representation of the Leech Lattice. ==Generalized Monstrous 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 groups, the relevant McKay–Thompson series is <math>T_{2A}(\tau)</math> = {1, 0, 276, {{val|fmt=commas|−2048}}, {{val|fmt=commas|11202}}, {{val|fmt=commas|−49152}}, ...} ({{OEIS2C|id=A007246}}) and <math>T_{4A}(\tau)</math> = {1, 0, 276, {{val|fmt=commas|2048}}, {{val|fmt=commas|11202}}, {{val|fmt=commas|49152}}, ...} ({{OEIS2C|id=A097340}}) where one can set the constant term {{nowrap|1=a(0) = 24}}, :<math>\begin{align} j_{4A}(\tau) &= T_{4A}(\tau) + 24 \\ &= \left(\frac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)}\right)^{24} \\ &= \left(\left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^4 + 4^2 \left(\frac{\eta(4\tau)}{\eta(\tau)}\right)^4\right)^2 \\ &= \frac{1}{q} + 24 + 276q + 2048q^2 + 11202q^3 + 49152q^4 + \dots \end{align}</math> and ''η''(''τ'') is the [[Dedekind eta function]]. == References == <references/> * {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | title=A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups | mr=0237634 | year=1968 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | volume=61 | pages=398–400 | doi=10.1073/pnas.61.2.398 | issue=2| pmc=225171 | pmid=16591697| bibcode=1968PNAS...61..398C | doi-access=free }} *{{Citation | editor1-last=Brauer | editor1-first=R. | editor1-link=Richard Brauer | editor2-last=Sah | editor2-first=Chih-han | title=Theory of finite groups: A symposium | url=https://books.google.com/books?id=YRdCAAAAIAAJ | publisher=W. A. Benjamin, Inc., New York-Amsterdam | mr=0240186 | year=1969}} *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | title=A group of order 8,315,553,613,086,720,000 | mr=0248216 | year=1969 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=1 | pages=79–88 | doi=10.1112/blms/1.1.79}} *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | editor1-last=Powell | editor1-first=M. B. | editor2-last=Higman | editor2-first=Graham | editor2-link=Graham Higman | title=Finite simple groups | url=https://books.google.com/books?id=TPPkAAAAIAAJ | publisher=[[Academic Press]] | location=Boston, MA | series=Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969. | isbn=978-0-12-563850-0 | mr=0338152 | year=1971 | chapter=Three lectures on exceptional groups | pages=215–247}} Reprinted in {{harvtxt|Conway|Sloane|1999|loc=267–298}} *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=Neil J. A. | author2-link=Neil Sloane | title=Sphere Packings, Lattices and Groups | url=https://books.google.com/books?id=upYwZ6cQumoC | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-98585-5 | mr=0920369 | year=1999 | volume=290 | doi=10.1007/978-1-4757-2016-7| url-access=subscription }} *{{Citation | last1=Thompson | first1=Thomas M. | title=From error-correcting codes through sphere packings to simple groups | url=https://books.google.com/books?id=ggqxuG31B3cC | publisher=[[Mathematical Association of America]] | series=Carus Mathematical Monographs | isbn=978-0-88385-023-7 | mr=749038 | year=1983 | volume=21}} *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Parker | first2=Richard A. | last3=Norton | first3=Simon P. | last4=Curtis | first4=R. T. | last5=Wilson | first5=Robert A. | title=Atlas of finite groups | url=https://books.google.com/books?id=38fEMl2-Fp8C | publisher=[[Oxford University Press]] | isbn=978-0-19-853199-9 | mr=827219 | year=1985}} * {{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}} * [https://web.archive.org/web/20051110090124/http://web.mat.bham.ac.uk/atlas/v2.0/spor/Co1/ Atlas of Finite Group Representations: Co<sub>1</sub>] version 2 * [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/ Atlas of Finite Group Representations: Co<sub>1</sub>] version 3 *{{Citation | last1=Wilson | first1=Robert A. | title=The maximal subgroups of Conway's group Co₁ | doi=10.1016/0021-8693(83)90122-9 | mr=723071 | year=1983 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=85 | issue=1 | pages=144–165| doi-access=free }} *{{Citation | last1=Wilson | first1=Robert A. | title=On the 3-local subgroups of Conway's group Co₁ | doi=10.1016/0021-8693(88)90192-5 | mr=928064 | year=1988 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=113 | issue=1 | pages=261–262| doi-access=free }} *{{Citation | last1=Wilson | first1=Robert A. | 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 | zbl=1203.20012 | year=2009| volume=251 }} *{{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=Collected papers. Gesammelte Abhandlungen | series=Springer Collected Works in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-57061-5 | mr=1643949 | year=1998 | doi=10.1007/978-3-642-41970-6}} *R. T. Curtis and B. T. Fairburn (2009), "Symmetric Representation of the elements of the Conway Group .0", Journal of Symbolic Computation, 44: 1044–1067. {{DEFAULTSORT:Conway Group}} [[Category:Sporadic groups]] [[Category:John Horton Conway]]
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