At 16:53 27/03/2002 +0000, Andrew Coppin wrote:
OK, not particularly fractal-related, but...
...can anyone remember the Cayley table for the "four group"? And while we're on the subject, how many distinct 6th-order groups are there? Obviously there's the 6th-order cyclic group... Then there's the product of C2 and C3... and I seem to recall the 3rd permutation group too... but is that isomorphic to C2 x C3? I'm not sure here... HEEELP!
As well as the cyclic group Z4, the other four-element group is Z2+Z2, which reflects complex arithmetic: * | e b c d --+--------- e | e b c d b | b e d c c | c d b e d | d c e b Substitute 1 for e, -1 for b, i for c and -i for d and the connection becomes obvious. There are two groups of order 6, Z6 and D3 - which are respectively Abelian and non-Abelian. Z6, the cyclic group of order 6, is simply addition modulo 6. It is isomorphic to the group Z2+Z3 (the product of the groups Z2 and Z3). The isomorphism runs as follows: Z2+Z3 <=> Z6 ------------ <0,0> <=> 0 <0,1> <=> 1 <0,2> <=> 2 <1,0> <=> 3 <1,1> <=> 4 <1,2> <=> 5 Rewrite the second column in base three and the relationship becomes obvious. The other group, D3, represents both the symmetries of an equilateral triangle and the permutations of three objects. If we say that the "three objects" are the vertices of such a triangle, the correspondence is again straightforward: Triangle Permutation symmetry ----------------------- 1 () 2 3 identity ------------ 2 (132) 3 1 120 degree rotation ------------ 3 (123) 1 2 240 degree rotation ------------ 1 (23) 3 2 vertical reflection ------------ 3 (13) 2 1 120 degree reflection ------------ 2 (12) 1 3 240 degree reflection Where a permutation is described in terms of which vertices have changed places; (123) says that "1 moves to 2's spot, 2 moves to 3's spot, and 3 moves to 1's spot" and (23) simply means that 2 and 3 have swapped places. The Cayley table for this group (and I'm going to use generic alphabetical labels) runs: * | e i j k l m --+------------ e | e i j k l m i | i j e l m k j | j e i m k l k | k m l e j i l | l k m i e j m | m l k j i e Note the lack of symmetry along the main diagonal as has been the case for Z2+Z2; this is a reflection of D3's non-Abelian nature - the operation denoted in the table by '*' is non-commutative. These correspond to permutations in the following way: e <=> () i <=> (132) j <=> (123) k <=> (23) l <=> (13) m <=> (12)
Thanks. Andrew.
PS. Appologies for wasting fractal bandwidth! (As if there's any other kind ;-)
You can make up for it by doing a Mandelbrot set for numbers built on some group other than Z2+Z2! Morgan L. Owens "There are 6 Abelian groups of order 1176."