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Janko group J4

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In the area of modern algebra known as group theory, the Janko group J4 is a sporadic simple group of order

   86,775,571,046,077,562,880
= 221 · 33 ··· 113 · 23 · 29 · 31 · 37 · 43
≈ 9×1019.

History[edit]

J4 is one of the 26 Sporadic groups. Zvonimir Janko found J4 in 1975 by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field with 2 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. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) gave a computer-free proof of existence by constructing it as an amalgams of groups 210:SL5(2) and (210:24:A8):2 over a group 210:24:A8.

The Schur multiplier and the outer automorphism group are both trivial.

Since 37 and 43 are not supersingular primes, J4 cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

Representations[edit]

The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.

The smallest permutation representation is on 173067389 points and has rank 20, with point stabilizer of the form 211:M24. The points can be identified with certain "special vectors" in the 112 dimensional representation.

Presentation[edit]

It has a presentation in terms of three generators a, b, and c as

Alternatively, one can start with the subgroup M24 and adjoin 3975 involutions, which are identified with the trios. By adding a certain relation, certain products of commuting involutions generate the binary Golay cocode, which extends to the maximal subgroup 211:M24. Bolt, Bray, and Curtis showed, using a computer, that adding just one more relation is sufficient to define J4.

Maximal subgroups[edit]

Kleidman & Wilson (1988) found the 13 conjugacy classes of maximal subgroups of J4 which are listed in the table below.

Maximal subgroups of J4
No. Structure Order Comments
1 211:M24 501,397,585,920
= 221·33·5·7·11·23
contains a Sylow 2-subgroup and a Sylow 3-subgroup; contains the centralizer 211:(M22:2) of involution of class 2B
2 21+12
+
·3.(M22:2)
21,799,895,040
= 221·33·5·7·11
centralizer of involution of class 2A; contains a Sylow 2-subgroup and a Sylow 3-subgroup
3 210:L5(2) 10,239,344,640
= 220·32·5·7·31
4 23+12·(S5 × L3(2)) 660,602,880
= 221·32·5·7
contains a Sylow 2-subgroup
5 U3(11):2 141,831,360
= 26·32·5·113·37
6 M22:2 887,040
= 28·32·5·7·11
7 111+2
+
:(5 × GL(2,3))
319,440
= 24·3·5·113
normalizer of a Sylow 11-subgroup
8 L2(32):5 163,680
= 25·3·5·11·31
9 PGL(2,23) 12,144
= 24·3·11·23
10 U3(3) 6,048
= 25·33·7
contains a Sylow 3-subgroup
11 29:28 812
= 22·7·29
Frobenius group; normalizer of a Sylow 29-subgroup
12 43:14 602
= 2·7·43
Frobenius group; normalizer of a Sylow 43-subgroup
13 37:12 444
= 22·3·37
Frobenius group; normalizer of a Sylow 37-subgroup

A Sylow 3-subgroup of J4 is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3.

References[edit]

  • Aschbacher, Michael; Segev, Yoav (1991), "The uniqueness of groups of type J4", Inventiones Mathematicae, 105 (3): 589–607, doi:10.1007/BF01232280, ISSN 0020-9910, MR 1117152, S2CID 121529060
  • D.J. Benson The simple group J4, PhD Thesis, Cambridge 1981, https://web.archive.org/web/20110610013308/http://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf
  • Bolt, Sean W.; Bray, John R.; Curtis, Robert T. (2007), "Symmetric Presentation of the Janko Group J4", Journal of the London Mathematical Society, 76 (3): 683–701, doi:10.1112/jlms/jdm086
  • Ivanov, A. A. (1992), "A presentation for J4", Proceedings of the London Mathematical Society, Third Series, 64 (2): 369–396, doi:10.1112/plms/s3-64.2.369, ISSN 0024-6115, MR 1143229
  • Ivanov, A. A.; Meierfrankenfeld, Ulrich (1999), "A computer-free construction of J4", Journal of Algebra, 219 (1): 113–172, doi:10.1006/jabr.1999.7851, ISSN 0021-8693, MR 1707666
  • Ivanov, A. A. (2004). The Fourth Janko Group. Oxford Mathematical Monographs. Oxford: Clarendon Press. 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 0931511
  • S. P. Norton The construction of J4 in The Santa Cruz conference on finite groups (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.

External links[edit]