a question on permutations in group theory.?

Question

Find the orders of the following members of (S)16 and determine which pairs of them are conjugate:
(a) (1 5 7 16) (2 11 13) (3 6) (8 10 12)
(b) (3 4) (5 9) (6 14) (7 10 11)
(c) (1 14) (2 4 8) (5 10) (6 7)

order of (a) is12,
order of (b) is 6
order of (c) is 6
i really don't understand how to find the conjugate of a permutation, its not the inverse so im not sure can sum1 who knows help plz.

Answer 1

Have a look at the link which is part of the Open University course in Pure Mathematics. The applet doesn't work for me but scroll down the text to find the section on conjugacy. Have a look at the solutions and compare with your own text books.

Basically the answer to your question is (b) and (c) since conjugacy preserves the cycle types. These both contain 1 3-cycle and 3 2-cycles.

You are right that conjugacy is not the same as the inverse. The inverse of an element g is the element that when combined with g gives the identity (if gh = e then h is the inverse of g).

g is conjugate to h in the group G if there is an element k in G satisfying g = khk^-1 (where k^-1 is the inverse of k)

Answer 2

there must be a description of the conjugate permutation,
i think they must have similar decomposition into cycles like b and c.