On the girth of random Cayley graphs

We prove that random d‐regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (logd‐1|G|)1/2/2 and that random d‐regular Cayley graphs of simple algebraic groups over 𝔽q asymptotically almost surely have girth at least log d‐1|G|/dim(G). For the symmetric p‐groups the girth is between loglog |G| and (log |G|)α with α < 1. Several conjectures and open questions are presented.