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Some examples of invariably generated groups

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A group G is invariably generated (IG) if there is a subset SG such that for every subset S′ ⊆ G, obtained from S by replacing each element with a conjugate, S′ generates G. Likewise, G is finitely invariably generated (FIG) if, in addition, one can choose such a subset S to be finite.

In this note we construct a FIG group G with an index 2 subgroup N < G such that N is not IG. This shows that neither property IG nor FIG is stable under passing to subgroups of finite index, answering questions of Wiegold and Kantor-Lubotzky-Shalev. We also produce first examples of finitely generated IG groups that are not FIG, answering a question of Cox.

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The author would like to thank Charles Cox whose talk on the paper [5] motivated this work, and who gave useful comments on a preliminary version of this article. The author also thanks the anonymous referee for many useful suggestions which led to improvements of the exposition.

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Correspondence to Ashot Minasyan.

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Minasyan, A. Some examples of invariably generated groups. Isr. J. Math. 245, 231–257 (2021).

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