When do random subsets decompose a finite group?

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Abstract

Let A, B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group, i.e. {AB ∪ BA = G}. The study of this event gives rise to a group invariant we call Θ(G). Θ(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes √Θ(G)|G| log |G|; i.e. for any < 0, if the size of A and B is less than (1 - )√Θ(G)|G| log |G|, then with high probability AB ∪ BA ≠ G. If A and B are larger than (1 + )√Θ(G)|G| log |G|, then AB ∪ BA = G with high probability.

Original languageEnglish
Pages (from-to)203-219
Number of pages17
JournalIsrael Journal of Mathematics
Volume174
Issue number1
DOIs
StatePublished - 1 Nov 2009
Externally publishedYes

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