Product of two prime cycles

**bulls6x** · January 13th 2009, 07:09 PM

Let $\sigma,\tau\in{S_p}$ be cycles of length p, where p is a prime. Prove or disprove that $\sigma\tau$ is a cycle of length p.

**NonCommAlg** · January 13th 2009, 08:27 PM

Originally Posted by bulls6x

Let $\sigma,\tau\in{S_p}$ be cycles of length p, where p is a prime. Prove or disprove that $\sigma\tau$ is a cycle of length p.

it's trivially false: for example if $\tau=\sigma^{-1},$ you'll get a counter-example. a better question is to see if we can find all cycles $\sigma, \tau \in S_n$ of length $n$ such that $\sigma \tau$ is also a cycle of length $n.$

another question (probably easier): is it always true that if $\sigma, \tau$ and $\sigma \tau$ are cycles of length $n$ in $S_n,$ then $\tau \sigma$ is also a cycle of length $n$ ?

**ThePerfectHacker** · January 13th 2009, 08:28 PM

Originally Posted by bulls6x

Let $\sigma,\tau\in{S_p}$ be cycles of length p, where p is a prime. Prove or disprove that $\sigma\tau$ is a cycle of length p.

Well if $\tau = \sigma^{-1}$ then $\sigma \tau = \text{ id}$ . Thus, I guess you want to say $\sigma\tau$ is a $p$ -cycle or the identity. However, that still doth not work because $(12345)(15234) = (24)(35)$ .

EDIT: NonCommAlg responded faster.

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