In the area of modern algebra known as group theory , the Janko group J 2 or the Hall-Janko group HJ is a sporadic simple group of order
30-524: [REDACTED] Look up hj in Wiktionary, the free dictionary. HJ may refer to: Science, technology, and mathematics [ edit ] Hall–Janko group , a mathematical group Hamilton–Jacobi equation , an equation in classical mechanics which also has relations to quantum physics U.S. code for a cryptographic key change; see cryptoperiod Other uses [ edit ] ⟨hj⟩ ,
60-652: A dodecad . Its centralizer has the form 2 :M 12 and has conjugates inside the monomial subgroup. Any matrix in this conjugacy class has trace 0. A permutation matrix of shape 2 1 can be shown to be conjugate to an octad ; it has trace 8. This and its negative (trace −8) have a common centralizer of the form (2 ×2).O 8 (2) , a subgroup maximal in Co 0 . Conway and Thompson found that four recently discovered sporadic simple groups, described in conference proceedings ( Brauer & Sah 1969 ), were isomorphic to subgroups or quotients of subgroups of Co 0 . Conway himself employed
90-485: A permutation representation of degree 100. It is also a subgroup of index two of the group of automorphisms of the Hall–Janko Near Octagon , leading to a permutation representation of degree 315. It has a modular representation of dimension six over the field of four elements; if in characteristic two we have w + w + 1 = 0 , then J 2 is generated by the two matrices and These matrices satisfy
120-471: A rank 3 permutation group on 100 points. Both the Schur multiplier and the outer automorphism group have order 2. As a permutation group on 100 points J 2 has involutions moving all 100 points and involutions moving just 80 points. The former involutions are products of 25 double transportions, an odd number, and hence lift to 4-elements in the double cover 2.A 100 . The double cover 2.J 2 occurs as
150-623: A subgroup of the Conway group Co 0 . J 2 is the only one of the 4 Janko groups that is a subquotient of the monster group ; it is thus part of what Robert Griess calls the Happy Family. Since it is also found in the Conway group Co1 , it is therefore part of the second generation of the Happy Family. It is a subgroup of index two of the group of automorphisms of the Hall–Janko graph , leading to
180-805: A complex representation of the Leech Lattice. 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 T 2 A ( τ ) {\displaystyle T_{2A}(\tau )} = {1, 0, 276, −2,048 , 11,202 , −49,152 , ...} ( OEIS : A007246 ) and T 4 A ( τ ) {\displaystyle T_{4A}(\tau )} = {1, 0, 276, 2,048 , 11,202 , 49,152 , ...} ( OEIS : A097340 ) where one can set
210-401: A group normal in a copy of S 3 , which commutes with a simple subgroup of order 168. A direct product PSL(2,7) × S 3 in M 24 permutes the octads of a trio and permutes 14 dodecad diagonal matrices in the monomial subgroup. In Co 0 this monomial normalizer 2 :PSL(2,7) × S 3 is expanded to a maximal subgroup of the form 2.A 9 × S 3 , where 2.A 9 is the double cover of
240-463: 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 1 . The inner product on the Leech lattice is defined as 1/8 the 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
270-468: A notation for stabilizers of points and subspaces where he prefixed a dot. Exceptional were .0 and .1 , being Co 0 and Co 1 . For integer 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
300-529: A two-letter combination used in some languages hj-reduction in English, dropping the / h / sound before / j / Hajji (Hj.), an Islamic honorific Handjob hic jacet ('here lies'), Latin phrase on gravestones Hilal-i-Jurat , post-nominal for Pakistan honour Hitler-Jugend (Hitler Youth) Holden HJ , an Australian car 1974-1976 Hot Jupiter , a type of planet Tasman Cargo Airlines , IATA airline designator Jiaguwen Heji ,
330-450: Is called a frame or cross . N has as an orbit a standard frame of 48 vectors of form (±8, 0 ). The subgroup fixing a given frame is a conjugate of N . The group 2 , isomorphic to the Golay code, acts as sign changes on vectors of the frame, while M 24 permutes the 24 pairs of the frame. Co 0 can be shown to be transitive on Λ 4 . Conway multiplied the order 2 |M 24 | of N by
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#1732852369703360-503: 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. Thomas Thompson ( 1983 ) relates how, in about 1964, 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
390-603: Is commonly called an h-k-l triangle . In the simplest cases Co 0 is transitive on the points or triangles in question and stabilizer groups are defined up to conjugacy. Conway identified .322 with the McLaughlin group McL (order 898,128,000 ) and .332 with the Higman–Sims group HS (order 44,352,000 ); both of these had recently been discovered. Here is a table of some sublattice groups: Two sporadic subgroups can be defined as quotients of stabilizers of structures on
420-517: Is different from Wikidata All article disambiguation pages All disambiguation pages Hall%E2%80%93Janko group J 2 is one of the 26 Sporadic groups and is also called Hall–Janko–Wales group . In 1969 Zvonimir Janko predicted J 2 as one of two new simple groups having 2 :A 5 as a centralizer of an involution (the other is the Janko group J3 ). It was constructed by Marshall Hall and David Wales ( 1968 ) as
450-453: Is only one conjugacy class of J 2 in G 2 (4). Every subgroup J 2 contained in G 2 (4) extends to a subgroup J 2 :2 = Aut(J 2 ) in G 2 (4):2 = Aut( G 2 (4)) ( G 2 (4) extended by the field automorphisms of F 4 ). G 2 (4) is in turn isomorphic to a subgroup of the Conway group Co 1 . There are 9 conjugacy classes of maximal subgroups of J 2 . Some are here described in terms of action on
480-569: Is the subgroup (2.A 7 × PSL 2 (7)):2 . Next comes (2.A 6 × SU 3 (3)):2 . The unitary group SU 3 (3) (order 6,048 ) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is (2.A 5 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 (2.A 4 o 2.G 2 (4)):2 , G 2 (4) being an exceptional group of Lie type . The chain ends with 6.Suz:2 (Suz= Suzuki sporadic group ), which, as mentioned above, respects
510-522: The Mathieu group M 24 (as permutation matrices ). N ≈ 2 :M 24 . A standard 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 sextet . The matrices of Co 0 are orthogonal ; i. e., they leave the inner product invariant. The inverse is the transpose . Co 0 has no matrices of determinant −1. The Leech lattice can easily be defined as
540-424: The Z - module generated by the set Λ 2 of all vectors of type 2, consisting of and their images under N . Λ 2 under N falls into 3 orbits of sizes 1104, 97152, and 98304 . Then | Λ 2 | = 196,560 = 2 ⋅3 ⋅5⋅7⋅13 . Conway strongly suspected that Co 0 was transitive on Λ 2 , and indeed he found a new matrix, not monomial and not an integer matrix. Let η be
570-494: The 4-by-4 matrix Now let ζ be a block sum of 6 matrices: odd numbers each of η and − η . ζ is a symmetric and orthogonal matrix, thus an involution . Some experimenting shows that it interchanges vectors between different orbits of N . To compute |Co 0 | it is best to consider Λ 4 , 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 { v , – v }. A set of 48 such vectors
600-469: The Conway groups, Co 0 , is the group of automorphisms of the Leech lattice Λ with respect to addition and inner product . It has order but it is not a simple group. The simple group Co 1 of order is defined as the quotient of Co 0 by its center , which consists of the scalar matrices ±1. The groups Co 2 of order and Co 3 of order consist of the automorphisms of Λ fixing
630-504: The Hall–Janko graph. The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall–Janko graph. Conway group#Suzuki chain of product groups In the area of modern algebra known as group theory , the Conway groups are the three sporadic simple groups Co 1 , Co 2 and Co 3 along with the related finite group Co 0 introduced by ( Conway 1968 , 1969 ). The largest of
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#1732852369703660-410: 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 Conway agreed to look at the problem. John G. Thompson said he would be interested if he were given the order of
690-535: The Leech lattice. Identifying R with C and Λ with the resulting automorphism group (i.e., the group of Leech lattice automorphisms preserving the complex structure ) when divided by the six-element group of complex scalar matrices, gives the Suzuki group Suz (order 448,345,497,600 ). This group was discovered by Michio Suzuki in 1968. A similar construction gives the Hall–Janko group J 2 (order 604,800 ) as
720-413: The alternating group A 9 . John Thompson pointed out it would be fruitful to investigate the normalizers of smaller subgroups of the form 2.A n ( Conway 1971 , p. 242). Several other maximal subgroups of Co 0 are found in this way. Moreover, two sporadic groups appear in the resulting chain. There is a subgroup 2.A 8 × S 4 , the only one of this chain not maximal in Co 0 . Next there
750-511: The equations (Note that matrix multiplication on a finite field of order 4 is defined slightly differently from ordinary matrix multiplication. See Finite field § Field with four elements for the specific addition and multiplication tables, with w the same as a and w the same as 1 + a .) J 2 is thus a Hurwitz group , a finite homomorphic image of the (2,3,7) triangle group . The matrix representation given above constitutes an embedding into Dickson's group G 2 (4) . There
780-448: The group. Conway expected to spend months or years on the problem, but found results in just a few sessions. Witt (1998 , page 329) stated that he found the Leech lattice in 1940 and hinted that he calculated the order of its automorphism group Co 0 . Conway started his investigation of Co 0 with a subgroup he called N , a holomorph of the (extended) binary Golay code (as diagonal matrices with 1 or −1 as diagonal elements) by
810-408: The number of frames, the latter being equal to the quotient | Λ 4 |/48 = 8,252,375 = 3 ⋅5 ⋅7⋅13 . That product is the order of any subgroup of Co 0 that properly contains N ; hence N is a maximal subgroup of Co 0 and contains 2-Sylow subgroups of Co 0 . N also is the subgroup in Co 0 of all matrices with integer components. Since Λ includes vectors of
840-565: The quotient of the group of quaternionic 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 . Co 0 has 4 conjugacy classes of elements of order 3. In M 24 an element of shape 3 generates
870-500: The shape (±8, 0 ) , Co 0 consists of rational matrices whose denominators are all divisors of 8. The smallest non-trivial representation of Co 0 over any field is the 24-dimensional one coming from the Leech lattice, and this is faithful over fields of characteristic other than 2. Any involution in Co 0 can be shown to be conjugate to an element of the Golay code. Co 0 has 4 conjugacy classes of involutions. A permutation matrix of shape 2 can be shown to be conjugate to
900-519: The standard comprehensive collection of rubbings of ancient Chinese oracle bone inscriptions Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title HJ . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=HJ&oldid=1214108582 " Category : Disambiguation pages Hidden categories: Short description
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