A class of rearrangement groups that are not invariably generated

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-04-26 DOI:10.1112/blms.13046

Davide Perego, Matteo Tarocchi

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Abstract

A group $G$ is invariably generated if there exists a subset $S \subseteq G$ such that, for every choice $g_{s} \in G$ for $s \in S$ , the group $G$ is generated by ${s^{g_{s}} ∣ s \in S}$ . Gelander, Golan, and Juschenko (J. Algebra 478 (2016), 261–270) showed that Thompson groups $T$ and $V$ are not invariably generated. Here, we generalize this result to the larger setting of rearrangement groups, proving that any subgroup of a rearrangement group that has a certain transitive property is not invariably generated.

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一类并非一成不变生成的重排基团

如果存在一个子集 S ⊆ G $S \subseteq G$，使得对于每一个选择 g s ∈ G $g_s \in G$ for s ∈ S $s \in S$，群 G $G$ 由 { s g s ∣ s∈ S } 生成，那么群 G $G$ 不变地生成。 $lbrace s^{g_s}\mid s \in S \rbrace$ .Gelander、Golan 和 Juschenko (J. Algebra 478 (2016), 261-270) 证明汤普森群 T $T$ 和 V $V$ 并非不变地生成。在此，我们将这一结果推广到更大的重排群环境中，证明重排群的任何子群，只要具有一定的传递性质，都不是不变生成的。

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