Let $\displaystyle \sigma,\tau\in{S_p}$ be cycles of length p, where p is a prime. Prove or disprove that $\displaystyle \sigma\tau$ is a cycle of length p.
it's trivially false: for example if $\displaystyle \tau=\sigma^{-1},$ you'll get a counter-example. a better question is to see if we can find all cycles $\displaystyle \sigma, \tau \in S_n$ of length $\displaystyle n$ such that $\displaystyle \sigma \tau$ is also a cycle of length $\displaystyle n.$
another question (probably easier): is it always true that if $\displaystyle \sigma, \tau$ and $\displaystyle \sigma \tau$ are cycles of length $\displaystyle n$ in $\displaystyle S_n,$ then $\displaystyle \tau \sigma$ is also a cycle of length $\displaystyle n$?
Well if $\displaystyle \tau = \sigma^{-1}$ then $\displaystyle \sigma \tau = \text{ id}$. Thus, I guess you want to say $\displaystyle \sigma\tau$ is a $\displaystyle p$-cycle or the identity. However, that still doth not work because $\displaystyle (12345)(15234) = (24)(35)$.
EDIT: NonCommAlg responded faster.