my-server
← Wiki

Janko group J3

In the area of modern algebra known as group theory, the Janko group J<sub>3</sub> or the Higman-Janko-McKay group HJM is a sporadic simple group of order

&nbsp;&nbsp;&nbsp;50,232,960 = 2<sup>7</sup>3<sup>5</sup>51719.

History and properties

J<sub>3</sub> is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 2<sup>1+4</sup>:A<sub>5</sub> as a centralizer of an involution (the other is the Janko group J<sub>2</sub>). J<sub>3</sub> was shown to exist by .

In 1982 R. L. Griess showed that J<sub>3</sub> cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

J<sub>3</sub> 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. 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. Two from the ATLAS list are 18×18 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 J<sub>3</sub>: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 J<sub>3</sub>:2. One then defines the following relation:

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

Curtis showed, using a computer, that this relation is sufficient to define J<sub>3</sub>:2.

Using a presentation

In terms of generators a, b, c, and d its automorphism group J<sub>3</sub>:2 can be presented as

A presentation for J<sub>3</sub> in terms of (different) generators a, b, c, d is

Maximal subgroups

found the 9 conjugacy classes of maximal subgroups of J<sub>3</sub> as follows:

References

  • R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p.&nbsp;93: proof that J<sub>3</sub> is a pariah.
  • Z. Janko, Some new finite simple groups of finite order, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp.&nbsp;25–64 Academic Press, London, and in The theory of finite groups (Edited by Brauer and Sah) p.&nbsp;63-64, Benjamin, 1969.

External links