113256 wrote:how do you raise a permutation group to a power. i know how to do it for small powers like 2 but how do you find T^387
T =
(1 2 3 4 5 6 7 8)
(3 4 5 2 7 8 1 6)
This permutation may be written in cyclic form as T = (1 3 5 7)(2 4)(6 8).
The order of this permutation is the least common multiple (LCM) of the lengths of these cycles; the lengths are 4, 2, and 2, so the order of T is 4. Because the permutations within T are disjoint, multiplication is associative and commutative. Thus, T^4 = (1 3 5 7)(2 4)(6 8)*(1 3 5 7)(2 4)(6 8)*(1 3 5 7)(2 4)(6 8)*(1 3 5 7)(2 4)(6 8) = (1 3 5 7)^4 * (2 4)^4 * (6 8)^4 = (1 3 5 7)(2 4)(6 8) = T.
Using this information, how might you reduce the power on T^387 to be a more manageable value?