Talk:Janko group J2

Involutions

The Hall-Janko group has 2 conjugacy classes of involutions. I want to find out how many points each moves. Then I will mention that in reference to the centralizers of involutions. So in the list of maximal subgroups there will soon be more than one reference to the Hall-Janko graph. Scott Tillinghast, Houston TX (talk) 04:24, 5 April 2008 (UTC)

In one class, each involution moves 80 points. In the other class, each involution moves 100 points. The 80 pointer has the centralizer of type 2^(4+1).A5, where the 2-group is extraspecial of order 2^5 and of quaternion type. The 100 pointer has the centralizer of type 2^2.A5, where the 2-group is elementary abelian.

Thanks for specifying the maximal subgroups and correcting/clarifying whether J2 contains A6 as a subgroup (it does not, but 3.PGL(2,9) contains the triple cover 3.A6 as an index 2 subgroup, and so J2 contains A6 as a nice subquotient). JackSchmidt (talk) 04:43, 5 April 2008 (UTC)

Thank you.

There are a few other matters I would like to include.

The simple group U₃(3) of order 6048 has orbits of 36 and 63 in J2. It permutes 36 simple subgroups of order 168. There are 2 conjugacy classes of 63 maximal subgroups of order 96, non-isomorphic. There are 63 involutions, all conjugate; in J2 they would have to move 80 points. Also U₃(3) acts on rays of a projective space over field F₉ in orbits of 28 and 63. So I would like to identify the right objects with the 63 vertices of the Hall-Janko graph.

It looks likely that there are multiple conjugate sets of A₅. Maybe with different involutions, 3-elements, etc. Scott Tillinghast, Houston TX (talk) 06:29, 5 April 2008 (UTC)

My error: I looked at some old files I had made on G6048 and found that the centralizer of an involution IS the stabilizer of one ray of the 63. Consider, for example the subgroup of matrices having zero in the top row and left column except at top left corner. These commute with the diagonal matrix with diagonal 1,-1,-1 and also fix the ray (x,0,0). This group has order 96. Scott Tillinghast, Houston TX (talk) 21:06, 7 April 2008 (UTC)

The other maximal subgroup of G6048 of order 96 has structure 4²:S₃ and contains at least 9 involutions: 3 permutation matrices, 3 diagonal matrices, and 3 products of these. Scott Tillinghast, Houston TX (talk) 20:27, 9 April 2008 (UTC)

Matrix representaion

HJ is a maximal subgroup of the exceptional Lie-type group G2(4) over the field F4. The generating matrices shown here look like part of that embedding. G2(4) is in turn a subgroup of the Suzuki sporadic group, which in turn is a subquotient in Conway 1. I want to verify that we have here a representaion in G2(4) and find sources on when this was discovered and by whom.

The group G2(2) has the simple group G6048, of order 6048, as a subgroup of index 2. G6048 is a maximal subgroup of HJ. I wonder whether it is the "real" subgroup of our matrix representaion, in other words, the restriction to matrices with elements in field F2. Scott Tillinghast, Houston TX (talk) 20:29, 28 March 2009 (UTC)

I verified that a representation as you describe exists: there is a copy of HJ containing G2(2)' and contained in G2(4). I am not sure of the best way to give the sourced version. Hopefully the online ATLAS's rep is such a rep, and that can give wikipedia sourcing, but haven't had a chance to check. As far as scholarly attribution, I haven't had a chance to check either. JackSchmidt (talk) 19:44, 29 March 2009 (UTC)

I think the 1968 paper by Hall and Wales has something about the matrix representation over field F4, but I have not recently seen this paper. The journal and paper are at Rice University, but I am somewhat disabled and do not go there often.

I notice in the Atlas that both HJ and G2(2) (i. e. U3(3):2) are maximal subgroups of G2(4). Scott Tillinghast, Houston TX (talk) 04:03, 31 March 2009 (UTC)

This is described in a very citable manner in:

• Wales, David B. (1969), "Generators of the Hall-Janko group as a subgroup of G₂(4)", Journal of Algebra, 13: 513–516, doi:10.1016/0021-8693(69)90113-6, ISSN 0021-8693, MR 0251133

The language is that of Chevalley groups, but is fairly clear. It shows that there are exactly two conjugacy classes of HJ in G2(4), and they are conjugate by the field automorphism. Their intersection (the "real" subgroup) is G2(2)'. Let me know if you want an electronic copy. JackSchmidt (talk) 06:04, 30 June 2009 (UTC)

Thank you very much. I think I can get access to this paper. Scott Tillinghast, Houston TX (talk) 03:49, 3 July 2009 (UTC)

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