The trivial lower bound for the girth of $S_n$

Abstract

Consider the Cayley graph of $S_n$ generated by a random pair of elements $x,y$. Conjecturally, the girth of this graph is $\Omega (n\log n)$ with probability tending to $1$ as $n\to \infty$. We show that it is at least $\Omega (n^{1/3})$.

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