# Math Help - Permutations, cycles

1. ## Permutations, cycles

Let t be an element of S be the cycle (1,2....k) of length k with k<=n.
a) prove that if a is an element of S then ata^-1=(a(1),a(2),...,a(k)). Thus ata^-1 is a cycle of length k.
b)let b be any cycle of length k. Prove there exists a permutation a an element of S such that ata^-1=b.

We assume t is an element of S and a is an element S.
By definition of elements of S if t is in S, we have a determined by t(1), t(2),...,t(n)
Furthermore if a is in S, we have a(1), a(2)....a(n).
That's as far as I get.

2. So I have ata^-1=at(n)
because a^-1 sends a(1)--->1,a(2)--->2,.....a(n)--->n
By defininition of t, we have a(1), a(2), a(3)...a(k). We go to k because our definition of t says k<=n.

For b,
Let b be any cycle of length k.
We have (1,2.....k). I'm not sure how to show the rest.