By Dan Segal
After a forty-year lull, the research of word-values in teams has sprung again into existence with a few unbelievable new leads to finite crew thought. those are principally prompted through functions to profinite teams, together with the answer of an previous challenge of Serre. This publication offers a accomplished account of the recognized effects, either previous and new. The extra uncomplicated tools are constructed from scratch, resulting in self-contained proofs and enhancements of a few vintage effects approximately limitless soluble teams. this can be via an in depth advent to extra complicated subject matters in finite staff concept, and a whole account of the functions to profinite teams. the writer provides proofs of a few very fresh effects and discusses open questions for additional learn. This self-contained account is obtainable to investigate scholars, yet will curiosity all examine employees in staff conception.
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Extra resources for Words: Notes on Verbal Width in Groups (London Mathematical Society Lecture Note Series)
M ⊆ Ew∗m . Thus w has width m in E. Therefore w(E1 ) = E1,w Remark. It may have occurred to the reader that similar results for nilpotent groups could have been established without using generalized words at all. Is it worth all that trouble to make the small extension to virtually nilpotent groups? This is a moot point, but the techniques will be used in an essential way in later sections. 2. 20 Chapter 2. 2 Group ring stuﬀ Here we collect some module-theoretic results needed in the following section.
3. 1. Let E be a ﬁnitely generated group and A1 an abelian normal subgroup with E/A1 virtually nilpotent. 2 there exists n such that wE (A1 ) ⊆ Ew∗n . Thus to show that w has ﬁnite width in E it will suﬃce to show that it has ﬁnite width in E/wE (A1 ). e. that A1 ≤ w ∗ (E). 3, there exist normal subgroups A ≤ H of E, with A ≤ A1 , such that E/H is ﬁnite, A = γc (H) and A ∩ Z(H) = 1. Thus G = H/A is ﬁnitely generated nilpotent and A is ﬁnitely generated as a G-module. 4, and let K0 denote the subset consisting of E-invariant members of K.
Then p F ∩ F ≤ (F )p (exercise! ). As F is free abelian of rank d(d − 1)/2 it follows that G is elementary abelian of the same rank, while G/Z(G) is elementary abelian of rank d. 2. Commutators in p-groups 45 values in G (note that [x, y n ] = [xn , y] and [y, x] = [x, y −1 ] in G); while |γ2 (G)| = pd(d−1)/2 . Hence 2(2d − 1) 4 h dimG (Gγ 2 ) < < . d(d − 1) d−1 The preceding lemma now gives the result. A sharper result can be obtained by linear algebra, without using ﬁniteness. The following exercise implies a lower bound of [d/2] for the width of γ2 in Gd,p , and also in some torsion-free nilpotent groups, such as Fd /γ3 (Fd ) and its Mal’cev completion.