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We prove that random d-regular Cayley graphs of the symmetric group asymp-totically almost surely have girth at least (log d−1 |G|) 1/2 /2 and that random d-regular Cay-ley graphs of simple algebraic groups over F q asymptotically almost surely have girth at least log d−1 |G|/ dim(G). For the symmetric p-groups the girth is between log log |G| and (log |G|)… (More)

We employ concepts and tools from the theory of finite permutation groups in order to analyse the Hidden Subgroup Problem via Quantum Fourier Sampling (QFS) for the symmetric group. We show that under very general conditions both the weak and the random-strong form (strong form with random choices of basis) of QFS fail to provide any advantage over… (More)

We assume that for some <i>fixed</i> large enough integer d, the symmetric group S<inf>d</inf> can be generated as an expander using d<sup>1/30</sup> generators. Under this assumption, we explicitly construct an infinite family of groups G<inf>n</inf>, and explicit sets of generators Y<inf>n</inf> ⊂ G<inf>n</inf>, such that all generating sets have… (More)

We construct a sequence of groups G n , and explicit sets of generators Y n ⊂ G n , such that all generating sets have bounded size, and the associated Cayley graphs are all expanders. The group G 1 is the alternating group A d , the set of even permutations on the elements {1, 2,. .. , d}. The group G n is the group of all even symmetries of the rooted… (More)

In the paper we pose and discuss new Burnside-type problems, where the role of nilpo-tency is replaced by that of solvability.

A theorem of O'Nan and Scott [6]; [2, Chapter 4] restricts the possibilities for maximal subgroups of finite symmetric groups: they are of six types of which the first four are explicitly known, the fifth involves a finite simple group, and the sixth an action of a simple group. This result, in conjunction with the Classification of Finite Simple Groups,… (More)

- Aner Shalev
- IJAC
- 1997

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