On Mon, Aug 27, 2012 at 2:26 PM, Tom Bachmann <[email protected]> wrote: > I'm not sure if I'm helping, but I'm also not sure if I understand what you > are saying. > > Let us fix a set X we are considering the permutation group of, below I will > take X = {1, 2, 3, 4, 5}. A permutation of X is by definition a bijective > function f:X->X. It is specified uniquely by providing the image of every > element. We can write this in the short form > [f(1), f(2), f(3), f(4), f(5)]. In this way, every permutation is > represented by an array of constant size. > > Now let's talk about cycles. By definition, a cycle is an ordered subset of > X. In general, the cycle (a_1 ... a_n) represents the unique permutation f > with f(a_1) = a_2, ..., f(a_n) = a_1, and f(x) = x for all x not in {a_1, > .., a_n}. For example, the cycle (1 2 3) denotes the permutation [2, 3, 1, > 4, 5]. > We can identify the set of cycles with a subset of the set of permutations. > > Now let's consider composition. There are two schools of thought. Let me > write * for "ordinary" (where I come from) composition, and . for "weird" > composition. By definition, if f, g are permutations, then the permutation > f*g is the unique mapping such that (f*g)(x) = f(g(x)) ["apply right to > left"], whereas (f.g)(x) = g(f(x)) ["apply left to right"]. [*] >
This is correct *as functions*. If you do this *as permutations* (which *act* on a set and are not just functions), then you want the permutations to form an "action" on the set. The way you want to do it, they aren't an action: if g1 = (1,2) and g2 = (2,4,5) then (g1*g2)(2) \not= g1(g2(2)): sage: G = SymmetricGroup(5) sage: g1 = G([(1,2)]) sage: g2 = G([(2,4,5)]) sage: g2(2) 4 sage: g3 = g1*g2 sage: a = g2(2); a 4 sage: b = g1(a); b 4 sage: g3(2) 1 It is very important for many people that the group of permutations yields a group action on the set. > Observe now that any permutation can be written uniquely (up to ordering) as ... > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to > [email protected]. > For more options, visit this group at > http://groups.google.com/group/sympy?hl=en. > -- You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
