论文标题
某些特殊方程式的满意度
Degree of satisfiability of some special equations
论文作者
论文摘要
Gustafson的一个著名定理指出，在一个非亚洲集团中，$ xy = yx $的满意度，即两个均匀随机选择的群体元素$ x，y $ ofere等式$ xy = yx $，不得大于$ \ frac {5} {8} {8} {8} $。 Antolin，Martino和Ventura（Arxiv：1511.07269）的开创性工作对有限生成的组的满意度推广程度，导致对其他方程式的Gustafson风格的性质的重新兴趣。最近获得了2英格尔和Metabelian身份的积极结果（Arxiv：1809.02997）。在这里，我们表明方程式$ xy^2 = y^2x $，$ xy^3 = y^3x $和$ xy = yx^{ - 1} $是1，或者不超过$ 1- \ varepsilon $，对于某些正常数$ \ varepsilon $。使用Antolin-Martino-Ventura形式主义，我们引入标准，以确定哪些方程在有限指数亚组中，如果它们具有正面的满足程度。我们推断出方程$ xy = yx^{ - 1} $和$ xy^2 = y^2x $没有此属性。
A well-known theorem of Gustafson states that in a non-Abelian group the degree of satisfiability of $xy=yx$, i.e. the probability that two uniformly randomly chosen group elements $x,y$ obey the equation $xy=yx$, is no larger than $\frac{5}{8}$. The seminal work of Antolin, Martino and Ventura (arXiv:1511.07269) on generalizing the degree of satisfiability to finitely generated groups led to renewed interest in Gustafson-style properties of other equations. Positive results have recently been obtained for the 2-Engel and metabelian identities (arXiv:1809.02997). Here we show that the degree of satisfiability of the equations $xy^2=y^2x$, $xy^3=y^3x$ and $xy=yx^{-1}$ is either 1, or no larger than $1-\varepsilon$ for some positive constant $\varepsilon$. Using the Antolin-Martino-Ventura formalism, we introduce criteria to identify which equations hold in a finite index subgroup precisely if they have positive degree of satisfiability. We deduce that the equations $xy=yx^{-1}$ and $xy^2=y^2x$ do not have this property.
