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Let $G$ be a unique product group, i.e., for any two finite subsets $A$ and$B$ of $G$ there exists $x\in G$ which can be uniquely expressed as a productof an element of $A$ and an element of $B$. We prove that, if $C$ is a finitesubset of $G$ containing the identity element such that $\langle C\rangle$ isnot 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 identityelement of a torsion-free group $G$ such that $|C| = 3$ and $\langle C\rangle$is not abelian, then for all subsets $B$ of $G$ with $|B|\geq 7$, $|BC|\geq |B|+ 5$. Moreover, if $\langle C\rangle$ is not isomorphic to the Klein bottlegroup, i.e., the group with the presentation $\langle x, y \;|\; xyx =y\rangle$, then for all subsets $B$ of G with $|B|\geq 5$, $|BC|\geq |B| + 5$.The support of an element $\alpha =\sum_{x\in G} \alpha_x x$ a group algebra$F[G]$ ($F$ is any field), denoted by $supp(\alpha)$, is the set $\{x\in G\;|\; \alpha_x \neq 0\}$. 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)|\geq10$. 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] toarbitrary fields, respectively.

Cardinality of product sets in torsion-free groups and applications in group algebras