Summary: Let \(G\) be a unique product group, i.e. for any two finite subsets \(A,B\) of \(G\), there exists \(x \in G\) 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 \(\langle C \rangle\) 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 \(\langle C \rangle\) 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 bottle group, i.e. the group with the presentation \(\langle x,y\mid xyx=y \rangle\), then for all subsets \(B\) of \(G\) with \(|B| \geq 5, |BC| \geq |B|+5\). The support of an element \(\alpha= \sum_{x \in G} \alpha_xx\) in a group algebra \(\mathbb{F}[G] (\mathbb{F}\) is any field), denoted by \(\text{supp}(\alpha)\), is the set \(\{x\in G| \alpha_x \neq 0\}\). By the latter result, we prove that if \(\alpha \beta = 0\) for some nonzero \(\alpha, \beta \in \mathbb{F}[G]\) such that \(|\text{supp}(\alpha)|=3\), then \(|\text{supp}(\beta)|\geq 12\). Also, we prove that if \(\alpha\beta=1\) for some \(\alpha,\beta\in\mathbb{F}[G]\) such that \(|\text{supp}(\alpha)|=3\), then \(|\text{supp}(\beta)|\geq 10\). These results improve a part of results in [P. Schweitzer, J. Group Theory 16, No. 5, 667–693 (2013; Zbl 1292.20007); K. Dykema et al., Exp. Math. 24, No. 3, 326–338 (2015; Zbl 1403.20004)] to arbitrary fields, respectively.