Abstract
Abstract Invariable generation is a topic that has predominantly been studied for finite groups. In 2014, Kantor, Lubotzky and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various authors have investigated the property for particular infinite groups or families of infinite groups. A group is invariably generated by a subset S if replacing each element of S with any of its conjugates still results in a generating set for G. In this paper, we investigate how this property behaves with respect to wreath products. Our main work is to deal with the case where the base of G ≀ X H {G\wr_{X}H} is not invariably generated. We see both positive and negative results here depending on H and its action on X.
Highlights
Invariable generation arises naturally in computational Galois theory and has been actively studied in relation to many interesting questions
The same authors worked with invariable generation for infinite groups, developing a wide range of results in [7]
In [11, 12] the notion of groups where no proper subgroup meets every conjugacy class was considered, which is equivalent to the group being invariably generated; [11] showed that this property is closed under extensions, whereas [12] showed that it is not always preserved for subgroups
Summary
Invariable generation arises naturally in computational Galois theory and has been actively studied in relation to many interesting questions. There are many exciting and unexpected results in this area, such as [6, Theorem 1.3] which says that any non-abelian finite simple group can be invariably generated by two elements. Other results in [7] do not add extensively to this If both G and H are FIG or IG, it follows from [7, Corollary 2.3 (iii)], which states that the class of invariably generated groups is closed under extensions, that G oX H is IG or FIG. If X is infinite and G is non-trivial, the base of G oX H will not be finitely generated and so cannot be FIG. This makes the other tools in [7] mostly unusable for studying wreath products.
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