Coessential Abelianization Morphisms in the Category of Groups

Abstract

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.

Recalcitrance in groups II

“…Since the abelianization homomorphism is coessential if G is finite (this follows from the so-called ''Gaschü tz Lemma''-or see [6]), we infer Corollary 2.1. If G is a finite group of rank n and principal length l, then recðGÞ c nðl À 1Þ þ 2.…”

“…From the Theorem of [6], giving conditions for a group to have coessential abelianization homomorphism, we immediately infer the following corollary. Corollary 3.1.…”

“…Theorem 2.2. Let G be any group of rank n with a principal series and for which the abelianization epimorphism is coessential (that is, any generating n-tuple of G ab can be naturally lifted to a generating n-tuple for G-see [6]). If l is the principal length of G, then recðGÞ c nðl À 1Þ þ 2.…”

On Andrews–Curtis conjectures for soluble groups

“…Proof. Since G is two-generated, the introductory remark of [24] implies that π ab is coessential if and only if every generating pair of G ab lifts to a generating pair of G. Let g be a generating pair of G ab . By [13, Corollary 1], there is a generating pair h which is Nielsen equivalent to g and of the form (u, a k ) for some u ∈ R × C and some k ∈ Z such that C = a k .…”