I am reading on symmetric groups (chapter 1 section 3 on dummit and foote) and I have noticed that, given an $m-$cycle, say $(a_1,a_2 \cdots a_m)$, If I compose $\sigma$ with itself, suppose $i-$times, depending on $gcd(i,m)$, the cycle either stays an $m$-cycle, or splits. For eg, for $m=10$, and $i=3$, we see that the cycle is still an $10-$cycle, but if $i=2$, the cycle splits into 2, 5 sized disjoint cycles. If $i=4$, then still its 2* 5 sized cycles, but each one is a rearrangement away

from being the same as what we get for $\sigma^2$. How do I understand what is happening here?

I don't understand the following:
The text defines: More generally, let $\chi : M \to M_0$ be a smooth map such that $\chi(p) = p_0$.
Any tangent vector $v \in T_p M$ gives rise to a tangent vector $D_p \chi(v) \in T_{p_0} M_0$ defined by
$(D_p \chi(v))(f) := v(f \circ \chi)$.

And then a page later, we are talking about the tangent bundle, and a maximal atlas defined upon it,
which contains all charts of the form $D\psi : T U \to V \times \mathbb{R}^n : (p, v) := (\psi(p), D_p\psi(v))$.
But obviously the second element is not in $\mathbb{R}^n$, it is a map?