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• W2950773072 abstract "Let $G$ be a unique product group, i.e., for any two finite subsets $A$ and $B$ of $G$ there exists $xin G$ 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 $langle Crangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|geq 7$, $|BC|geq |B| + |C| + 2$. 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 $langle Crangle$ is not abelian, then for all subsets $B$ of $G$ with $|B|geq 7$, $|BC|geq |B| + 5$. Moreover, if $langle Crangle$ is not isomorphic to the Klein bottle group, i.e., the group with the presentation $langle x, y ;|; xyx = yrangle$, then for all subsets $B$ of G with $|B|geq 5$, $|BC|geq |B| + 5$. The support of an element $alpha =sum_{xin G} alpha_x x$ a group algebra $F[G]$ ($F$ is any field), denoted by $supp(alpha)$, is the set ${xin G ;|; alpha_x neq 0}$. By the latter result, we prove that if $alpha beta = 0$ for some non-zero $alpha,beta in F[G]$ such that $|supp(alpha)| = 3$, then $|supp(beta)|geq 12$. Also, we prove that if $alpha beta= 1$ for some $alpha beta in F[G]$ such that $|supp(alpha)| = 3$, then |supp(alpha)|geq 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." @default.
• W2950773072 created "2019-06-27" @default.
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• W2950773072 date "2018-08-27" @default.
• W2950773072 modified "2023-09-27" @default.
• W2950773072 title "Cardinality of product sets in torsion-free groups and applications in group algebras" @default.
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