Residually nilpotent one-relator groups with nontrivial centre

Abstract

Abstract. We determine explicitly the residually nilpotent one-relator groups with nontrivial centre. We show also that, if G is a one-relator group, then G is residually nilpotent if, and only if, its central quotient G/Z(G) is residually nilpotent.In this note we shall present an explicit determination of those one-relator groups with nontrivial centre that are residually nilpotent. The proof depends on the structure theory of one-relator groups with nontrivial centre developed by A. Pietrowski in [P].In studying the residual nilpotence of groups which tend to be highly noncommutative, a concept which we call "compactness" is of some significance. A grouptheoretic class X is termed compact if every noncyclic and residually nilpotent group in X is residually a finite p-group, for some prime number p. The importance of this property results from the fact that the class of residually p-finite groups is much better behaved than the class of residually nilpotent groups. For example, a free product of residually p-finite groups is residually p-finite, while the free product of a group of order two and a group of order three is not residually nilpotent. Incidentally, this reflects the fact that the class of nontrivial free products of finite or cyclic groups is compact. Thus, if we wish to prove that a group is residually nilpotent, it is helpful to know that we may as well attempt to prove that it is residually a finite p-group, for some prime number p. In particular, the presence of torsion renders this approach exceedingly useful.In [M], it was observed that the class of one-relator groups with torsion is compact. This raises the question whether the class of all one-relator groups is compact. As a first step towards dealing with torsion free one-relator groups, we sought to determine whether the class of one-relator groups with nontrivial centre is compact. Not only were we able to confirm this, but our investigation led to a complete and explicit determination of all residually nilpotent one-relator groups with nontrivial centre.Here is our main result.

Гомотопические И Комбинаторные Аспекты Теории Нормальных Рядов В Группах

“…Мы имеем группу , являющуюся группой с одним соотношением и нетривиальным центом. Из работы [30] следует, что может быть представлена с помощью одного из следу-ющих копредставлений:…”

Untitled

“…Мы имеем группу , являющуюся группой с одним соотношением и нетривиальным центом. Из работы [30] следует, что может быть представлена с помощью одного из следующих копредставлений: Доказательство. Пусть заданы свободными копредставлениями…”

Baer invariants and residual nilpotence of groups

“…Доказательство. По теореме 2, примененной к k-центральному расширению (10), имеем N ∩ γ m ( G) = 0, откуда следует, что для любого l m существует естественный изоморфизм γ l ( G) = γ l (G), из которого непосредственно вытекает требуемое утверждение.…”