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Subgroup
Normal subgroup Quotient group Group homomorphism (Semi-)direct product group homomorphisms, kernel, image, direct sum, wreath product, simple, finite, infinite, continuous, multiplicative, additive, cyclic, abelian, dihedral, nilpotent, solvable, List of group theory topics Glossary of group theory |
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Classification of finite simple groups Mathieu groups
M11, M12, M22, M23, M24 Conway groups Co1, Co2, Co3 |
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Integers Z |
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Solenoid Infinite dimensional Lie group
O(∞), SU(∞), Sp(∞) |
In mathematics, the fourth Janko group J4 is the sporadic finite simple group of order 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 = 86775571046077562880 whose existence was suggested by Zvonimir Janko (1976). Its existence and uniqueness was shown by Simon P. Norton and others in 1980. Janko found it by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). It has a modular representation of dimension 112 over the finite field of two elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. The Schur multiplier and the outer automorphism group are both trivial. Ivanov (2004) has given a proof of existence and uniqueness that does not rely on computer calculations.
J4 is one of the 6 sporadic simple groups called the pariahs, because they are not found within the Monster group. The order of the monster group is not divisible by 37 or 43.
Presentation [link]
It has a presentation in terms of three generators a, b, and c as
- Failed to parse (Missing texvc executable; please see math/README to configure.): a^2=b^3=c^2=(ab)^{23}=[a,b]^{12}=[a,bab]^5=[c,a]=
- Failed to parse (Missing texvc executable; please see math/README to configure.): (ababab^{-1})^3(abab^{-1}ab^{-1})^3=(ab(abab^{-1})^3)^4=
- Failed to parse (Missing texvc executable; please see math/README to configure.): [c,bab(ab^{-1})^2(ab)^3]=(bc^{bab^{-1}abab^{-1}a})^3=
- Failed to parse (Missing texvc executable; please see math/README to configure.): ((bababab)^3cc^{(ab)^3b(ab)^6b})^2=1.
Maximal subgroups [link]
Kleidman & Wilson (1988) showed that J4 has 13 conjugacy classes of maximal subgroups.
- 211:M24 - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 211:(M22:2), centralizer of involution of class 2B
- 21+12.3.(M22:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups
- 210:PSL(5,2)
- 23+12.(S5 × PSL(3,2)) - containing Sylow 2-subgroups
- U3(11):2
- M22:2
- 111+2:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
- PSL(2,32):5
- PGL(2,23)
- U3(3) - containing Sylow 3-subgroups
- 29:28 = F812
- 43:14 = F602
- 37:12 = F444
A Sylow 3-subgroup is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3
References [link]
- D.J. Benson The simple group J4, PhD Thesis, Cambridge 1981, https://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf
- Ivanov, A. A. The fourth Janko group. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004. xvi+233 pp. ISBN 0-19-852759-4 MR2124803
- Z. Janko, A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroups, J. Algebra 42 (1976) 564-596.doi:10.1016/0021-8693(76)90115-0 (The title of this paper is incorrect, as the full covering group of M22 was later discovered to be larger: center of order 12, not 6.)
- Kleidman, Peter B.; Wilson, Robert A. (1988), "The maximal subgroups of J4", Proceedings of the London Mathematical Society. Third Series 56 (3): 484–510, DOI:10.1112/plms/s3-56.3.484, ISSN 0024-6115, MR 931511
- S. P. Norton The construction of J4 in The Santa Cruz conference on finite groups (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.
- Atlas of Finite Group Representations: J4
https://wn.com/Janko_group_J4
Janko group
In the area of modern algebra known as group theory, the Janko groups are the four sporadic simple groups J1, J2, J3 and J4 introduced by Zvonimir Janko. Unlike the Mathieu groups, Conway groups, or Fischer groups, the Janko groups do not form a series, and the relation among the four groups is mainly historical rather than mathematical.
History
Janko constructed the first of these groups, J1, in 1965 and predicted the existence of J2 and J3. In 1976, he suggested the existence of J4. Later, J2, J3 and J4 were all shown to exist.
J1 was the first sporadic simple group discovered in nearly a century: until then only the Mathieu groups were known, M11 and M12 having been found in 1861, and M22, M23 and M24 in 1873. The discovery of J1 caused a great "sensation" and "surprise" among group theory specialists. This began the modern theory of sporadic groups.
And in a sense, J4 ended it. It would be the last sporadic group (and, since the non-sporadic families had already been found, the last finite simple group) predicted and discovered, though this could only be said in hindsight when the Classification theorem was completed.
