2018/08/27 by Abdollahi, Alireza, Jafari, Fatemeh
#16S34 #20C05 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #Primary: 20D60 #Rings and Algebras (math.RA) #Secondary: 11P70
paper · doi:10.48550/arxiv.1808.08708
Let G be a unique product group, i.e., for any two finite subsets A and B of G there exists x∈ G which can be uniquely expressed as a product of an element of A and an element of B. We prove that, if C is a finite subset of G containing the identity element such that ⟨ C⟩ is not abelian, then for all subsets B of G with |B|≥ 7, |BC|≥ |B| + |C| + 2. Also, we prove that if C is a finite subset containing the identity element of a torsion-free group G such that |C| = 3 and ⟨ C⟩ is not abelian, then for all subsets B of G with |B|≥ 7, |BC|≥ |B| + 5. Moreover, if ⟨ C⟩ is not isomorphic to the Klein bottle group, i.e., the group with the presentation ⟨ x, y | xyx = y⟩, then for all subsets B of G with |B|≥ 5, |BC|≥ |B| + 5. The support of an element α=∑x∈ G αx x a group algebra F[G] (F is any field), denoted by supp(α), is the set \x∈ G | αx ≠ 0\. By the latter result, we prove that if αβ= 0 for some non-zero α,β∈ F[G] such that |supp(α)| = 3, then |supp(β)|≥ 12. Also, we prove that if αβ= 1 for some αβ∈ F[G] such that |supp(α)| = 3, then |supp(α)|≥ 10. These results improve a part of results in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667-693] and Dykema et al. [Exp. Math., 24 (2015), 326-338] to arbitrary fields, respectively.