Abstract. Motivated by the well-known conjecture of Andrews and Curtis [1], we consider the question of how, in a given n-generator group G, any ordered n-tuple of ''annihilators'' of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitrance of G is defined to be the least number of elementary standard moves (''elementary M-transformations'') by means of which every annihilating n-tuple can be transformed into a generating n-tuple. We obtain upper estimates for the recalcitrance of n-generator finite groups-thus quantifying a result from [2]-and of a wide class of n-generator solvable groups, thus extending and correcting a result from [3].
Nielsen transformations determine the automorphisms of a free group of rank n, and also of a free abelian group of rank n, and furthermore the generating n-tuples of such groups form a single Nielsen equivalence class. For an arbitrary rank n group, the generating n-tuples may fall into several Nielsen classes. Diaconis and Graham ['The graph of generating sets of an abelian group', Colloq. Math. 80 (1999), 31-38] determined the Nielsen classes for finite abelian groups. We extend their result to the case of infinite abelian groups.2010 Mathematics subject classification: primary 20F99; secondary 20K99.
Abstract. An epimorphism φ : G → H of groups, where G has rank n, is called coessential if every (ordered) generating n-tuple of H can be lifted along φ to a generating n-tuple for G. We discuss this property in the context of the category of groups, and establish a criterion for such a group G to have the property that its abelianization epimorphism G → G/ [G, G], where [G, G] is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews-Curtis conjecture.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.