无挠的群中积集的基数及其在群代数中的应用
Cardinality of product sets in torsion-free groups and applications in group algebras
摘要：
假设$G$是一个独特的产品组，即，对于$G$的任意两个有限子集$A$和$B$，存在$x\in G$，它可以唯一地表示为$A$的一个元素和$B$的一个元素的乘积.我们证明了，如果$C$是$G$的一个有限子集，包含单位元使得$\langle C\rangle$不是阿贝尔的，那么对于$G$的所有子集$B$与$|B|\geq 7$，$|BC|\geq |B| + |C| + 2$ .证明了：若$C$是一个包含无挠的群$G$的单位元的有限子集，使得$|C| = 3$和$\langle C\rangle$不是阿贝尔群，则对于$G$的所有子集$B$和$|B|\geq 7$，$|BC|\geq |B| + 5$ .此外，如果$\langle C\rangle$不同构于克莱因瓶群，即，表示为$\langle x, y \;|\; xyx = y\rangle$的组，则对于G的所有子集$B$具有$|B|\geq 5$，$|BC|\geq |B| + 5$。群代数$F[G]$（$F$是任何域）的元素$α=\sum_{x\in G} α_x x$的支集，记为$supp(α)$，是集合$\{x\in G \;|\; α_x \neq 0\}$。利用后一个结果，我们证明了：如果$αβ= 0$对于某个非零的$α,β\in F[G]$使得$|supp(α)| = 3$，则$|supp(β)|\geq 12$ .此外，我们证明了，如果$αβ= 1$对于某些$αβ\in F[G]$，使得$|supp(α)| = 3$，则|补充，补充|\geq 10$.这些结果改进了Schweitzer [J. Group Theory，16（2013），no.5，667-693]和Dykema等人[Exp.数学、24（2015），326-338]分别到任意字段。
英文摘要：
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 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 $\langle x, y \;|\; xyx = y\rangle$, then for all subsets $B$ of G with $|B|\geq 5$, $|BC|\geq |B| + 5$. The support of an element $α=\sum_{x\in G} α_x x$ a group algebra $F[G]$ ($F$ is any field), denoted by $supp(α)$, is the set $\{x\in G \;|\; α_x \neq 0\}$. By the latter result, we prove that if $αβ= 0$ for some non-zero $α,β\in F[G]$ such that $|supp(α)| = 3$, then $|supp(β)|\geq 12$. Also, we prove that if $αβ= 1$ for some $αβ\in F[G]$ such that $|supp(α)| = 3$, then |supp(α)|\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.
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