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Subnormal Structure of Finite Soluble Groups

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Subnormal Structure of Finite Soluble Groups Subnormal Structure of Finite Soluble Groups Chris Wetherell September 2001 A thesis submitted for the degree of Doctor of Philosophy of the Australian National University For Alice, Winston and Ammama Declaration The work in this thesis is my own except where otherwise stated. Chris Wetherell Acknowledgements I am greatly indebted to my supervisor Dr John Cossey, without whose guidance and expertise this work would not have been possible. It need hardly be said that my understanding and appreciation of the subject are entirely due to his influence; I can only hope that I have managed to return the favour in some small way. I would also like to thank my two advisors: Dr Elizabeth Ormerod for her help with p-groups, and Professor Mike Newman for his general advice. I am particularly grateful to Dr Bob Bryce who supervised me during one of John's longer absences. My studies at the Australian National University have been made all the more enjoyable by my various office-mates, Charles, David, Axel and Paola. I thank them for their advice, suggestions, tolerance and friendship. Indeed everyone in the School of Mathematical Sciences has helped to provide a stimulating and friendly environment. In particular I thank the administrative staff who have assisted me during my time here. Finally, it is with deepest gratitude that I acknowledge the support of my family, my friends and, above all, my dear wife Alice. Without her patience, understanding and love I would be lost. vii Abstract The Wielandt subgroup, the intersection of normalizers of subnormal subgroups, is non-trivial in any finite group and thus gives rise to a series whose length is a measure of the complexity of a group's subnormal structure. Another measure, akin to the nilpotency class of nilpotent groups, arises from the strong Wielandt subgroup, the intersection of centralizers of nilpotent subnormal sections. This thesis begins an investigation into how these two invariants relate in finite soluble groups. Complete results are obtained for metabelian groups of odd order: the strong Wielandt length of such a group is at most one more than its Wielandt length, and this bound is best possible. Some progress is made in the wider class of groups with p-length 1 for all primes p. A conjecture for all finite soluble groups, which may be regarded as a subnormal analogue of the embedding of the Kern, is also considered. ix Contents Acknowledgements vii Abstract ix Notation and terminology xiii 1 Introduction 1 1.1 Summary of main results . 2 1.2 Notes for the reader . 4 2 Preliminaries 5 2.1 Subgroup functions . 5 2.2 Miscellaneous results . 6 2.3 The Wielandt and strong Wielandt subgroups . 13 2.4 Constructions . 22 3 A conjecture 29 4 A lower strong Wielandt series 39 4.1 Series determined by subgroup functions . 39 4.2 Subdirect subgroup functions . 43 4.3 Examples . 44 4.4 The ! ∗-series . 46 5 Bounds on strong Wielandt length 51 5.1 A-groups . 51 5.2 Nilpotent groups . 52 5.3 Reduction lemmas . 55 5.4 Metabelian groups . 60 xi xii CONTENTS 5.5 Abelian-by-nilpotent groups . 62 5.6 Groups of p-length one for all p ..................... 64 5.7 Class of Sylow subgroups . 71 5.8 Strong Wielandt length of subgroups . 72 6 Splitting of nilpotent residuals 77 7 Further research 81 7.1 Conjecture A . 81 7.2 A weaker strong Wielandt subgroup . 82 7.3 Wielandt and strong Wielandt length . 85 A Cited results 87 Bibliography 95 Notation and terminology In the following G is a group and H; Hi;K are subgroups of G; • g; gi; h; k are elements of G; • X; Y are classes of groups; • ' is a homomorphism; • α is a subgroup function; • p is a prime, q is a power of a prime and π; πi are sets of primes; • n is a natural number. • Notation H 6 G, H < G H is a subgroup, proper subgroup of G H ¡ G, H ¡¡ GH is a normal subgroupy, subnormal subgroupy of G H ¡ G, H ¡ GH is minimal normaly, maximal normal in G · · HK the set hk h H; k K f j 2 2 g H o K the semidirect product of H and K, with H ¡HK and H K = 1 \ H K the direct product of H and K, with H; K ¡ HK and H K = 1 × \ H K the regular wreath product of H and K, with base group isomor- o phic to H H × · · · × K | j{zj } th πn projection onto the n factor of a direct product, given by πn :(g1; g2;:::) gn 7! ynot necessarily proper xiii xiv NOTATION AND TERMINOLOGY α(G H) the pre-image in G of α(G=H), where H ¡ G ÷ 1 if n = 0 αn(G) ( α(G αn 1(G)) if n > 1 ÷ − h 1 g h− gh, the conjugate of g by h 1 1 1 h [g; h] g− h− gh = g− g , the commutator of g and h [g1; : : : ; gn][[g1; : : : ; gn 1]; gn] for n 3 − > [H; K] [h; k] h H; k K h j 2 2 i [H1;:::;Hn][[H1;:::;Hn 1];Hn] for n 3 − > [H; nK][H; K; : : : ; K] n | {z } G0 [G; G], the derived subgroup of G G if n = 0 G(n) ( , terms of the derived series of G (n 1) (G − )0 if n > 1 G if n = 1 γn(G) ( , terms of the lower central series of G [γn 1(G);G] if n 2 − > ζ(G) the centre of G ζ (G) n1=1 ζn(G), the hypercentre of G 1 S GX, GN the X-residual (if it exists), nilpotent residual of G (smallest nor- mal subgroup with X-quotient, nilpotent quotient) GX, Fit G the X-radical (if it exists), Fitting subgroup of G (largest normal X-subgroup, nilpotent subgroup) c(G), d(G), l(G) the nilpotency class, derived length, nilpotent length of G (length of shortest series with central, abelian, nilpotent factors) CG(H) the centralizer of H in G CG(H=K) the pre-image in G of CG=K (H=K), where K ¡ G xv NG(H) the normalizer of H in G N NG(H)CNG(H)(H=H ), the strong normalizer of H in G κ(G) the Kern of G (intersection of normalizers of all subgroups) !(G) the Wielandt subgroup of G (intersection of normalizers of sub- normal subgroups) !(G) the strong Wielandt subgroup of G (intersection of strong nor- malizers of subnormal subgroups) p ! (G) the p-Wielandt subgroup of G (intersection of normalizers of p0- perfect subnormal subgroups) ! p(G) the p-strong Wielandt subgroup of G (intersection of strong nor- malizers of p0-perfect subnormal subgroups) wl(G), wl(G) the Wielandt length, strong Wielandt length of G (least n such that !n(G) = G, !n(G) = G) Frat G the Frattini subgroup of G (intersection of maximal subgroups) Cosoc G the cosocle of G (intersection of maximal normal subgroups) Soc G the socle of G (product of minimal normal subgroups) CoreG(H) the core of H in G (largest subgroup of H which is normal in G) HG the normal closure of H in G (smallest normal subgroup of G which contains H) G if n = 0 HG;n ( HG;n 1 H − if n > 1 s(G : H) the (subnormal) defect of H in G (for H ¡¡ G, the least n such that HG;n = H) g the order of g (least positive n such that gn = 1) j j G the order of G (number of elements in G) j j G : H G = H , the index of H in G j j j j j j xvi NOTATION AND TERMINOLOGY π(G) the set of prime divisors of G j j p0, π0 the set of primes not in p , π f g Sylp(G) the set of Sylow p-subgroups of G Hallπ(G) the set of Hall π-subgroups of G Gp;Gπ a Sylow p-subgroup, Hall π-subgroup of G Oπ(G) the π-residual of G (smallest normal subgroup with π-quotient) π1 πn πn π1 πn 1 O ··· (G)O (O ··· − (G)) Oπ(G) the π-radical of G (largest normal π-subgroup of G) Oπ1 πn (G)Oπn (G Oπ1 πn 1 (G)) ··· ÷ ··· − Aut G the group of automorphisms of G Paut G the group of power automorphisms of G (those ' Aut G such 2 that for all g G, g' = gn for some integer n depending on g) 2 A, N, S, E, G the class of finite abelian groups, finite nilpotent groups, finite soluble groups, all finite groups, all groups XY the class of groups which have a normal X-subgroup with Y- quotient X Y the class of groups which are the direct product of an X-group × and a Y-group Xπ the class of π-groups in X Xπ1 πn Xπ1 πn 1 Xπn ··· ··· − Fq the field with q elements Cn the cyclic group of order n 2 2 2 V4 the Klein 4-group, with presentation x; y x = y = (xy) = 1 h j i Sn the symmetric group of degree n; 3 2 2 S3 has presentation a; b a = b = (ab) = 1 , and h j a b a i b S4 = V4 o S3 is x; y; a; b x = y; x = y; y = xy; y = x ∼ j xvii An the alternating group of degree n; A4 = x; y; a as a subgroup of S4 h i D2n the dihedral group of order 2n, with presentation r; l rn = l2 = (rl)2 = 1 h j i Q8 the quaternion group of order 8, with presentation 4 2 2 j 1 i; j i = 1; j = i ; i = i− h j i GLn(q) the general linear group of degree n over Fq; a 3 a b b GL2(3) = Q8 o S3 is i; j; a; b i = j ; j = ij; i = j; j = i ∼ j SLn(q) the special linear group of degree n over Fq; SL2(3) = i; j; a as a subgroup of GL2(3) h i 3 Ep for p odd, the extra-special group of order p and exponent p, with presentation u; v up = vp = [u; v; u] = [u; v; v] = 1 h j i Terminology G is a π-group π(G) π ⊆ G is π-perfect Oπ(G) = G pp0 G is p-nilpotent Op0p(G) = G, or equivalently O (G) = 1 π0ππ0 G has π-length 1 Oπ0ππ0 (G) = G, or equivalently O (G) = 1 G is an X-group G X 2 G is a T -group all subnormal subgroups of G are normal G is Dedekind all subgroups of G are normal G is Hamiltonian G is Dedekind and non-abelian G is an A-group all Sylow subgroups of G are abelian G is monolithic G has a unique minimal normal subgroup (the monolith of G) G is single-headed G has a unique maximal normal subgroup xviii NOTATION AND TERMINOLOGY N G is perfect G = G0, or equivalently G = G G is elementary abelian G = Cp ::: Cp for some n and p ∼ × × n | {z } G is n-generator G can be generated by n elements, but no smaller set of elements generates G H and K permute HK = KH or equivalently HK 6 G section of G H=K with K ¡ H 6 G composition factor of G simple section H=K with H ¡¡ G chief factor of G section H=K with K ¡ G and H=K ¡ G=K · ' is universal ' Paut G and there exists an integer n with g' =
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