Question

Let T be an m-cycle ( 1 2 3 ... m ) , show that T^i is an m-cycle iff m,i are relativity prime

Answer 1

if there is more than a prove , plz , add it

Answer 2

I think I have part of the proof:

I started with a few small cases, for example, suppose m=5.
T=(12345)
T^2=(13524)
T^3=(14253)
T^4=(15432)

Let T=(123⋯m) be an m-cycle
Notice that for each 1≤k≤m, T^i: k→(k+i) mod m
Suppose T^i is an m-cycle,
then T^i=(1 (1+i)mod m (1+2i)mod m ⋯ (1+(m−1)i) mod m)
Furthermore, for 1≤n≤m−1, 1+ni≢1(mod m)
Thus ni≢0(mod m)
So m and i are relatively prime.