Talk:Janko group

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Would it be better to break this article up into four separate articles? Gene Ward Smith 22:01, 28 Oct 2004 (UTC)

[edit] Revisions on Janko 1 and Janko 2 groups

The presentations (i. e. generators and relations) taken from Robert A. Wilson's Atlas of Finite Groups are rather counterintuitive, and I doubt they have much value in an introductory article. I have therefore removed them from the sections on J₁ and J₂. I have added other information.

I find it intersting, however, that J₁ and J₂ both have generators a, b such that a²=b³=(ab)⁷=1. These relations alone define an infinite group, the symmetry group of a tiling of the hyperbolic plane. Assigning order 4 to the commutator [a,b] = ababb defines the simple group of order 168. For J₁and J₂, [a,b] has order 19 and 12 respectively.

The generators of the Hall-Janko group J₂ are written with several components equal to the cube of w, but w is a cube root of unity. I have changed the cubes to 1.

The removed presentations were:

For J₁: presentation in terms of two generators a and b and c = abab^-1 as

a² = b³ = (ab)⁷ = (abc³)⁵ = (abc⁶abab(ab ^{− 1})²)² = 1.

For J₂: in terms of two generators a and b as a² = b³ = (ab)⁷ = (ababab ^{− 1}ababab ^{− 1}abab ^{− 1}ab ^{− 1})³ = 1, in terms of which it has an outer automorphism sending b to b².

Scott Tillinghast, Houston TX 01:33, 20 February 2007 (UTC)