On Thu, 1 Nov 2007, William Stein wrote:
>
> On 11/1/07, [EMAIL PROTECTED] <[EMAIL PROTECTED]> wrote:
>>> In particular, how does applying (1,2,3)(4,5) to [0,1,2,3,4]
>>> result in what you claim, i.e., in [1, 2, 0, 4, 3]?
>>> It should either be an error, or maybe:
>>> [2,0,1,3,4]
>>> if say (1,2,3)(4,5) sends the "first entry", i.e., 0th position
>>> to the 1st, the 1st to 2nd, etc,
>>
>> That's what is going on -- I interpret the cycle notation as:
>>
>> 1 |--> 2
>> 2 |--> 3
>> 3 |--> 1
>> 4 |--> 5
>> 5 |--> 4
>>
>> and I don't know how you arrived at [2,0,1,3,4] -- that looks like (3,2,1)
>> to me.
>
> OK, consider the permutation (1,2,3), which permutes
> things as follows:
>
> 1 |--> 2
> 2 |--> 3
> 3 |--> 1
>
> Thus if the input is v = [a,b,c], then
> the output is [c,a,b]. I.e., the permutation sends
> the entry at position 1 to position 2, i.e., the a gets
> stuck in the middle, the entry at position 2 to position 3,
> i.e., the b goes to where c is, and the entry at position 3
> goes to the front, i.e., to where a is. Thus
> applied to [0,1,2], we get [2,0,1].
>
> Applying the permutation (1,2,3)(4,5) to [0,1,2,3,4]
> should either move the entry in position 1 (which happens
> to be called "0") to position 2, so that the output looks like
> [*,0,*,*,*]
> or if we do some weird 1-based thing, it would always
> fix the first entry, then send the entry in position 1 to
> position 2 and look like:
> [0,*,1,*,*]
> and neither of these are what you claim, i.e.,
> [1, 2, 0, 4, 3].
Your mapping looks backwards to me -- it seems like you're applying an inverse
map, which I find disturbing. If I understand correctly, you'd want:
sage: G = PermutationGroup([[(1,2)],[(1,2,3,4,5)]])
sage: g = G([2,3,1,5,4])
sage: h = G([(1,2,3),(4,5)])
sage: h == g
True
sage: v = g([1,2,3,4,5]); v
[3,1,2,5,4] #!!!!!
sage: G(v) == g
False
Under the way I've implemented this, the action on the list [1,...,n] is
trivially isomorphic to the group structure. You seem to be using a
left-action which doesn't really go well with the notation g(l).
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