Janko group J3

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In the area of modern algebra known as group theory, the Janko groupJ3 or the Higman-Janko-McKay groupHJM is a sporadic simple group of order

Contents

   27 ·35 ·5 ·17 ·19 = 50232960.

History and properties

J3 is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group J2). J3 was shown to exist by GrahamHigman and John McKay  ( 1969 ).

In 1982 R. L. Griess showed that J3 cannot be a subquotient of the monster group. [1] Thus it is one of the 6 sporadic groups called the pariahs.

J3 has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. Weiss (1982) constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements. It has a complex projective representation of dimension eighteen.

Constructions

Using matrices

J3 can be constructed by many different generators. [2] Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:

and

Using the subgroup PSL(2,16)

The automorphism group J3:2 can be constructed by starting with the subgroup PSL(2,16):4 and adjoining 120 involutions, which are identified with the Sylow 17-subgroups. Note that these 120 involutions are outer elements of J3:2. One then defines the following relation:

where is the Frobenius automorphism or order 4, and is the unique 17-cycle that sends

Curtis showed, using a computer, that this relation is sufficient to define J3:2. [3]

Using a presentation

In terms of generators a, b, c, and d its automorphism group J3:2 can be presented as

A presentation for J3 in terms of (different) generators a, b, c, d is

Maximal subgroups

Finkelstein & Rudvalis (1974) found the 9 conjugacy classes of maximal subgroups of J3 as follows:

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References

  1. Griess (1982): p. 93: proof that J3 is a pariah.
  2. ATLAS page on J3
  3. Bradley, J.D.; Curtis, R.T. (2006), "Symmetric Generationand existence of J3:2, the automorphism group of the third Janko group", Journal of Algebra, 304 (1): 256–270, doi: 10.1016/j.jalgebra.2005.09.046