论文标题
不变的一代和霍顿小组
Invariable generation and the Houghton groups
论文作者
论文摘要
霍顿组$ h_1，h_2，\ ldots $是一个无限组的家族。 1975年，Wiegold表明$ H_3 $始终产生（IG），但$ H_1 \ le H_3 $不是。一个自然的问题是，组$ h_2，h_3，\ ldots $是否全部IG。 Wiegold还说，在他发现的一个IG组的示例中，该组的子组不是IG，该子组从来没有有限的指数。另一个自然的问题是，$ h_3 $中是否有一个有限索引子组，不是IG。在本说明中，我们证明，对于\ {2，3，\ ldots \} $中的每个$ n \，$ h_n $及其所有有限索引子组都是Ig。 Minasyan和Goffer Lazarovich在2020年6月的独立工作非常好：他们表明，IG组可以具有不是IG的有限索引子组。
The Houghton groups $H_1, H_2, \ldots$ are a family of infinite groups. In 1975 Wiegold showed that $H_3$ was invariably generated (IG) but $H_1\le H_3$ was not. A natural question is then whether the groups $H_2, H_3, \ldots$ are all IG. Wiegold also ends by saying that, in the examples he had found of an IG group with a subgroup that is not IG, the subgroup was never of finite index. Another natural question is then whether there is a subgroup of finite index in $H_3$ that is not IG. In this note we prove, for each $n\in \{2, 3, \ldots\}$, that $H_n$ and all of its finite index subgroups are IG. The independent work of Minasyan and Goffer-Lazarovich in June 2020 frames this note quite nicely: they showed that an IG group can have a finite index subgroup that is not IG.
