Sumsets in dicyclic groups Q 4 n and Um,n

Abstract

Let G be an arbitrary group. For any subsets A and B of G, let A * B = { a * b ; a ∈ A , b ∈ B } where ` * ’ is the binary operation on G. By μ G ( r , s ) , we denote the minimum cardinality of the set A * B , where | A | = r and | B | = s . In 2003, Eliahou et al. proposed a conjecture that for every finite group G of order g and every pair of integers r , s with 1 ≤ r , s ≤ g , k G ( r , s ) ≤ μ G ( r , s ) , where k G ( r , s ) = min d ∈ H ( G ) { ( ⌈ r d ⌉ + ⌈ s d ⌉ − 1 ) d } and H ( G ) is the set of orders of finite subgroups of G. In 2003, Eliahou et al. verified the conjecture for dihedral group D q of index q = p v . In this paper, we determine the upper bound for the size of μ Q 4 n ( r , s ) and μ U m , n ( r , s ) , where Q 4 n = 〈 a , b : a 2 n = e , b 2 = a n , a b = b a − 1 〉 is a dicyclic group of order 4n and U m , n = 〈 a , b : a 2 n = e , b m = e , a b a − 1 = b − 1 〉 and verify the conjecture proposed by Eliahou et al. [7] for Q 4 n for n to be a prime power.