> ...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!

 

The direct product of C2 and C3 is cyclic. (This always happens with a direct

product of cycles whose orders are pairwise relatively prime. In this case,

consider the powers of the entry (1,1) to see that this element has order 6.)

The only other group of order 6 is the non-abelian dihedral group -- the set of

rotations and reflections that keep an equilateral triangle invariant in the plane.

This is actually the full symmetric group on three objects -- all permutations of

three things, under composition of permutations.

 

The table for the four group is:

 

   1    a    b    ab

   a    1    ab  b
   b    ab  1    a

   ab  b    a    1

 

ObFractal: when's a really truly truecolor capable 32-bit Fractint gonna be available? :)