TY - JOUR

T1 - When do random subsets decompose a finite group?

AU - Yadin, Ariel

PY - 2009/11/1

Y1 - 2009/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=74349109628&partnerID=8YFLogxK

U2 - 10.1007/s11856-009-0110-1

DO - 10.1007/s11856-009-0110-1

M3 - Article

AN - SCOPUS:74349109628

VL - 174

SP - 203

EP - 219

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -