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If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text. By contrast, it was shown recently by Pemantle, Peres, and Rivin that four random elements do invariably generate S n with probability bounded away from zero. We include a proof of this statement which, while sharing many features with their argument, is short and completely combinatorial. Source Duke Math.
Symmetric group on a finite set is 2-generated - Groupprops
Zentralblatt MATH identifier Keywords symmetric group invariable generation random generators. Invariable generation of the symmetric group.
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Duke Math. The motivation is that a quasigroup a structure whose Cayley table is a Latin square has non-trivial character theory if and only if its multiplication group is not 2-transitive — this is a special and rare event. Yet it happens for all groups of order greater than 2. Groups are indeed special! What about extensions to the infinite? Let us not be too ambitious, and begin with the countably infinite. Now the symmetric group on a countably infinite set is uncountable, and so cannot be generated by two elements; the finitary symmetric group is locally finite, and so for a different reason cannot be generated by two elements.
Also, there is no natural measure on the symmetric group. Yet it exists, and it was Dixon himself who discovered it in Although there is no natural measure on the symmetric group, there is a natural topology. Suppose that the points on which the group acts are the natural numbers. This definition of distance makes the symmetric group into a complete metric space.
ISBN 13: 9780521857215
If we omitted the clause involving the inverses, we would define a metric, but it would not be complete. A set is residual if it contains a countable intersection of open dense sets.
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Residual sets resemble sets of full measure in various ways; they are necessarily non-empty the Baire category theorem and they are closed under countable intersections; moreover, they have non-empty intersection with any non-empty open set. A permutation group on an infinite set is said to be highly transitive if it is n -tranitive that is, can map any n -tuple of distinct points to any other for all natural numbers n. It is not hard to see that a permutation group is highly transitive if and only if its closure is the symmetric group.
Now Dixon showed in For a residual set of pairs g,h of elements of the symmetric group on a countable set, the group generated by g and h is free and highly transitive. This resolves in a very strong way the question of the existence of 2-generator free highly transitive groups, first proved by Tom McDonough in For a last analogy I turn from groups to monoids, or what is almost the same thing automata.
A submonoid of T n is called synchronizing if it contains a function of rank 1 one whose image has cardinality 1. After a talk I gave on synchronization, Brendan McKay asked me about the probability that r random elements of T n generate a synchronizing monoid. For example, it is not hard to see that T 3 contains exactly 7 maximal non-synchronizing monoids, each of order 6 one of them is the symmetric group S 3 ; the sizes of their intersections can be computed, and so using inclusion-exclusion, an exact formula for the probability that k random elements generate a synchronizing monoid can be calculated.
Duke Mathematical Journal
Previous Next. I try to keep this series focussed on the symmetric groups, but it means not going down so many interesting byways! You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email.
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Skip to content. Before sketching the proof I will make some comments. Now Dixon showed in For a residual set of pairs g,h of elements of the symmetric group on a countable set, the group generated by g and h is free and highly transitive. Like this: Like Loading