Words: Notes on Verbal Width in Groups (London Mathematical Society Lecture Note Series)

Forty three. 2 Commutators in p-groups . . . . . . . . . . . . . . . . . . . . . . forty four four phrases and profinite teams four. 1 Verbal subgroups in profinite teams . . four. 2 Open subgroups . . . . . . . . . . . . . four. three Pronilpotent teams . . . . . . . . . . . four. four sort stuff . . . . . . . . . . . . . . . . four. five phrases of infinite width in pro-p teams . four. 6 Finite uncomplicated teams . . . . . . . . . . . four. 7 The Nikolov-Segal theorems . . . . . . . four. eight Uniformly elliptic phrases . . . . . . . . . four. nine On J-varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . forty seven forty seven fifty two fifty six sixty three sixty seven 70 seventy four 87 ninety one five Algebraic and analytic teams ninety six five. 1 Algebraic teams . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety six five. 2 Adelic teams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety eight vii viii Contents five. three five. four five. five On just-infinite profinite teams . . . . . . . . . . . . . . . . . . . 104 p-adic analytic teams . . . . . . . . . . . . . . . . . . . . . . . . 106 Analytic stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one hundred ten Appendix: the issues 113 Bibliography 116 Index one hundred twenty Preface ‘Something outdated, whatever new, whatever borrowed ... ’ certainly one of Frobenius’s many strong principles used to be to take advantage of personality concept to check the answer of equations in teams. for instance, he gave a formulation for the variety of suggestions in a finite staff G of equations like [x, y] = g (1) (here g ∈ G is a given aspect and the unknowns x, y diversity over G). After a protracted hole, those tools have been taken up and utilized with devastating effect to finite easy teams; fresh highlights are Shalev’s evidence that for any non-trivial staff be aware w, each portion of each sufficiently huge finite easy workforce is a made of 3 w-values, and the evidence (by Liebeck, O’Brien, Shalev and Tiep) of ‘Ore’s Conjecture’: Equation (1) is solvable for each aspect g in each finite basic workforce (of path ‘simple’ the following ability ‘non-abelian simple’). This – very fascinating – tale has to be informed in one other ebook, even though i'm going to survey the various ends up in §4. 6. my very own curiosity in phrases begun with Serre’s recognized evidence that during a finitely generated pro-p staff, each subgroup of finite index is open. This rests on a (fairly basic) outcome as a result of Peter Stroud: in a d-generator nilpotent workforce, each component to the derived crew is a fabricated from d commutators. A simply algebraic truth, this holds for teams of any dimension; yet for Serre’s theorem one merely applies it to finite p-groups. attempting to generalize Serre’s theorem to prosoluble teams, I needed to determine a end result like Stroud’s that will carry in all finite soluble teams, and the single means i'll find to do that used to be to exploit finiteness in a powerful approach (counting arguments). The restrict to finite teams grew to become extra evidently inevitable while Nikolay Nikolov and that i finally prolonged Serre’s theorem to all finitely generated profinite teams: the facts relies crucially at the classification of finite basic teams. It was once additionally transparent that on relocating from soluble teams to finite teams usually we'd need to contemplate workforce phrases except commutators. We have been hence ended in questions of the subsequent variety: given a (suitable) staff observe w, is there a functionality f on N such that the width of w in any finite crew G is bounded by way of f (d(G))?