# A doubt about Theorem 14 in textbook Algebra by Saunders MacLane and Garrett Birkhoff

I'm reading the proof of Theorem 14 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.

Any permutation $$\sigma \neq 1$$ on a finite set $$X$$ is a composite $$\gamma_{1} \cdot \cdot \gamma_{k}$$ of disioint cyclic permutations $$\gamma_{i}$$, each of length $$2$$ or more. Except for changes in the order of the cyclic factors, $$\sigma$$ has only one such decomposition.

They said that

Each of the points $${\sigma}^{i} {x}$$ in this set $$C$$ has the same orbit [under $$\sigma$$].

IMHO, this is only when $${\sigma}^{i} {x}$$ is a generator of $$C$$, i.e. $$\gcd(i,m) = 1$$. As such, I feel that the statement may be not correct.

Could you please verify my observation?

• He means that the orbit is under the permutation $\sigma$, not under the permutation $\sigma^i$ – fhn Jul 1 '20 at 15:55

If $$y=\sigma^i(x)$$, $$x=\sigma^{-i}(y)$$, we have $$\sigma^p=Id$$ implies that $$\sigma^{-i}=\sigma^{p-i}$$ and $$x=\sigma^{p-i}(y)$$.