At 20:27 02/04/2002 -0500, Multiple Bogeys wrote:
...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?
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
Yeah ... what he said. Morgan L. Owens "Ah, but how many of order 1176 are _non_-Abelian?"