Sean Eberhard

Normal covering numbers for groups and connections to additive combinatorics

The normal covering number \gamma(G) of a finite group G is the minimal size of a collection of proper subgroups whose conjugates cover the group. This definition is motivated by number theory and related to the concept of intersective polynomials. For the symmetric and alternating groups we will see how these numbers are closely connected to some elementary (as in "relating to basic concepts", not "easy") problems in additive combinatorics, and we will use this connection to better understand the asymptotics of \gamma(S_n) and \gamma(A_n) as n tends to infinity.