Lower Central and Dimension Series of Groups
A fundamental object of study in group theory is the lower central series of groups. Understanding its relationship with the dimension series, which consists of the subgroups determined by the augmentation powers, is a challenging task. This monograph pre
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1952
Roman Mikhailov
· Inder Bir Singh Passi
Lower Central and Dimension Series of Groups
ABC
Roman Mikhailov
Inder Bir S. Passi
Steklov Mathematical Institute Department of Algebra Gubkina 8 Moscow 119991 Russia [email protected]
Centre for Advanced Study in Mathematics Panjab University Chandigarh 160014 India [email protected]
ISBN: 978-3-540-85817-1 e-ISBN: 978-3-540-85818-8 DOI: 10.1007/978-3-540-85818-8 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008934461 Mathematics Subject Classification (2000): 18G10, 18G30, 20C05, 20C07, 20E26, 20F05, 20F14, 20F18, 20J05, 55Q40, c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
To Olga and Surinder
Preface
A fundamental object of study in the theory of groups is the lower central series of groups whose terms are defined for a group G inductively by setting γ1 (G) = G,
γn+1 (G) = [G, γn (G)]
(n ≥ 1),
where, for subsets H, K of G, [H, K] denotes the subgroup of G generated by the commutators [h, k] := h−1 k −1 hk for h ∈ H and k ∈ K. The lower central series of free groups was first investigated by Magnus [Mag35]. To recall Magnus’s work, let F be a free group with basis {xi }i∈I and A = Z[[Xi | i ∈ I]] the ring of formal power series in the non-commuting variables {Xi }i∈I over the ring Z of integers. Let U(A) be the group of units of A. The map xi → 1 + Xi , i ∈ I, extends to a homomorphism θ : F → U(A),
(1)
since 1 + Xi is invertible in A with 1 − Xi + Xi2 − · · · as its inverse. The homomorphism θ is, in fact, a monomorphism (Theorem 5.6 in [Mag66]). For a ∈ A, let an denote its homogeneous component of degree n, so that a = a0 + a1 + · · · + an + · · · . Define Dn (F ) := {f ∈ F | θ(f )i = 0, 1 ≤ i < n},
n ≥ 1.
It is easy to see that Dn (F ) is a normal subgroup of F and the series {Dn (F )}n≥1 is a central series in F , i.e., [F, Dn (F )] ⊆ Dn+1 (F ) for all n ≥ 1. Clearly, the intersection of the series {Dn (F )}n≥1 is trivial. Since {Dn (F )}n≥1 vii
viii
Preface
is a central series, we have γn (F ) ⊆ Dn (F ) for all n ≥ 1. Thus, it
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