number of transpositions and the composition

Question

Let $S_{n}$ symetric groups and $p$ is number of transpostion.

for all $\sigma \in S_{n}$ $$\sigma^{p} = \underbrace{\sigma\circ \sigma \circ\cdots\circ \sigma}_{p\ \text{times}} = Id $$

Is this result always valid ?

Answer 1

Take $(123) \in S_3$ then we can write $(123) = (13)(12)$ i.e it is the product of two transpositions, but $(123)^2 =(132) \neq 1$.