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THE FRATTINI SUBGROUP of a GROUP Este Artıculo Está Dedicado Con Todo Cari˜No a La Memoria De Chicho. Al No Formar Parte De T

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THE FRATTINI SUBGROUP of a GROUP Este Artıculo Está Dedicado Con Todo Cari˜No a La Memoria De Chicho. Al No Formar Parte De T MARGARITA MATHEMATICA EN MEMORIA DE JOSE´ JAVIER (CHICHO) GUADALUPE HERNANDEZ´ (Luis Espa˜nol yJuan L. Varona, editores), Servicio de Publicaciones, Universidad de La Rioja, Logro˜no, Spain, 2001. THE FRATTINI SUBGROUP OF A GROUP JAVIER OTAL Abstract. In this paper, we review the behaviour of the Frattini subgroup Φ(G) and the Frattini factor group G/Φ(G) of an infinite group G having in mind the most standard results of the finite case. Este art´ıculo est´a dedicado con todo cari˜no a la memoria de Chicho. Al no formar parte de tu grupo de investigaci´on, no tuve la fortuna de relacionar- me contigo por este motivo, pero s´ı que tuve la oportunidad de participar en algunos de los m´ultiples seminarios que organizabas. Por la naturaleza de mi especialidad y las caracter´ısticas de las personas a los que aquellos se dirig´ıan, siempre me suger´ıas temas que pudieran ser bien comprendidos por la audiencia, una preocupaci´on que te caracteriz´osiempre.Yoquer´ıa hablar (suavemente, claro) de aspectos de Grupos yde Curvas El´ıpticas, ante lo que siempre pon´ıas el grito en el cielo. Pero te hice trampas yconsegu´ı hablar delante de ti algo de tales curvas, con la excusa de referir el teorema de Fermat ydescribir unas parametrizaciones con la circunferencia. Y te gust´o, ¿recuerdas? Pero nunca te pude contar nada de mi especialidad ni de sus aplicaciones yes una pena que llevar´e siempre conmigo. Lamento de coraz´on que ahora ya no me puedas poner trabas. 1. Introduction In studying the behaviour of the maximal subgroups of a finite group G, Giovanni Frattini [11] introduced what he called the Φ-subgroup of G, the intersection of the maximal subgroups of G. Since then, this subgroup is usuallyknown as the Frattini subgroup of G and can be defined as follows. Given a group G, bydefinition, the Frattini subgroup of G is Φ(G)= {M ≤ G | M maximal in G}, provided G has maximal subgroups. Otherwise Φ(G)=G. In order to set up clearlythe contents of this survey,we begin with finite groups G. The main result of Frattini is Theorem A. If G is finite then Φ(G) is nilpotent. His proof is verysimple and is a fairlyapplication of what he knows as the Frattini argument and set off the importance of the normalityof maximal subgroups of G,a propertywhich characterizes the nilpotencyof G. On the other hand, Φ(G)isknown 2000 Mathematics Subject Classification. Primary 20E28; Secondary 20F18. Key words and phrases. Maximal subgroups, chief factors, Frattini subgroup, generalized nilpo- tent groups. 1 2JAVIEROTAL to be the set of non generators of the group G, that is, the set of elements of G which can be removed in anygenerator system of G. As an immediate consequence, the Frattini factor group G/Φ(G)ofG is cyclic if and only if G is cyclic. Gathering now, among other properties (see [17]), nilpotencyof the Frattini subgroup can be characterized bythe following ones. Theorem B. If G is a finite group then the following are equivalent: (1) G is nilpotent. (2) G/Φ(G) is nilpotent. (3) Everymaximal subgroup of G is normal in G. (4) G ≤ Φ(G). We mention that these (and other) properties do not characterize nilpotencyof infinite groups; see [28]. The main goal of this note is to give an small surveyon the current state of these concepts and related ideas. Throughout, our group-theoretic notation is standard and is taken from [28]. For the reader’s convenience we briefly recall here some of it. • If A abelian, the torsion-free rank r0(A)ofA is the cardinal of a maximal set of elements of infinite order, and if p is a prime its p-rank rp(A)isthecardinal of a maximal set of non-trivial elements of order a power of p.Therank of A is r(A)=r0(A)+ p r0(A) (see Fuchs [12]). These concepts have been naturally extended to arbitrarygroups. We mention that a group G is said to have finite abelian section rank if everyabelian section of G is has finite rank, for p =0 or a prime. A soluble group G has finite abelian section rank if an onlyif there exists a abelian subnormal series 1 = G0 G1 ··· Gn = G such that each rp(Gi+1/Gi) < ∞ (p ≥ 0). • If X is a class of groups, a poly-X-group is a group G with a finite series between 1 and G with factors X-groups. Similarlyis defined a poly-X -group if X is a theoretical propertyof groups. For example, a soluble group is a poly-abelian group while a nilpotent group is a poly-central group. On the other hand, a minimax group is a poly-(Min or Max) group. If we replace finite series by ascending series we obtain hyper- instead of poly-. G • AgroupG is said to be an XC-group if G/CG(x ) ∈ X for all x ∈ G.Best examples of this are FC, PC, CC,andMC-groups X = finite, polycyclic-by-finite, Chernikov and minimax groups); see [31], [10], [1], [19]. 2. Chief factors and maximal subgroups In infinite groups when dealing with the Frattini subgroup, the first difficulty which appears is the non existence of maximal subgroups. Example 1. Let p be a prime. Then the complex pn-roots of unity( n ≥ 0)forma | p group G = x1,x2, ... x1 =1,xi = xi−1,i>1 . In this example, G is periodic, non finitelygenerated, with proper subgroups cyclic and without maximal subgroups. G is called aPr¨ufer p-group. THE FRATTINI SUBGROUP OF A GROUP 3 Example 2. The wreath product G of a Pr¨ufer p-group and a Pr¨ufer q-group (p = q) is not locallynilpotent without maximal subgroups. The structure of the minimal normal subgroups M of a group G Is known provided some local information is available and we it (see Section B of [18]). Theorem 2.1. Let U be a chief factor of a group G. (1) If G is locallysoluble, then U is abelian. (2) If G is locally polycyclic, then U is abelian. (3) If G is locallysupersoluble, then U is cyclic of prime order. (4) If G is locallynilpotent, then U is central of prime order. Even in the case of locallyfinite groups, verylittle it is known about general chief factors. In manycases the maximal subgroups of certain groups G use to have finite index provided the chief factors of G are finite. In 1968, Robinson [26] (Theorems C and D) achieved a result of this kind. Theorem 2.2. The chief factors of a hyperabelian group with finite abelian section rank are finite and its maximal subgroups have finite index. If G is an FC-group, each normal closure x G is finite, and so a similar result holds for these groups. The same is true for PC and CC-groups ([10] and [1]) and more generallyfor MC-groups ([19]). Theorem 2.3. Let G be an MC-group. Then each chief factor of G is finite and each maximal subgroup of G has finite index. In this direction, Wehrfritz proved (see [33], Corollary4.28) Theorem 2.4. The chief factors of a soluble-by-finite subgroup of a finitely gener- ated linear group are finite and the maximal subgroups have finite index. In what it concerns to the normalityof the maximal subgroups, for manyyears, the best result was due to McLain. Theorem 2.5. A maximal subgroup of a locallynilpotent group is normal. As a consequence of an influential result of Robinson [27] characterizing the nilpo- tencyof a finitelygenerated group bythat of its images, we have Theorem 2.6. A finitelygenerated hyper-(abelian or finite) group is nilpotent if and onlyif each of its maximal subgroups is normal. Recently, Franciosi and de Giovanni [8] have shown the following. Theorem 2.7. Let G be a soluble residuallyfinite minimax group. Then G is finite-by-nilpotent if and onlyif it has finitelymanymaximal subgroups which are not normal in G. Corollary 2.7.1. If G is a soluble minimax group with finitelymanynon-normal maximal subgroups, then G is Chernikov-by-nilpotent. 4JAVIEROTAL 3. The nilpotency of the Frattini subgroup In contrast with the celebrated result of Frattini quoted in the Introduction, Just in the beginning of Section 2, we have seen an example of an infinite group G in which Φ(G) is neither locallynilpotent. Certainly,this group G is neither finitely generated nor has maximal subgroups, and these seem the basic reasons whythe nilpotencyof Φ( G) can fail. In his pioneer work on polycyclic groups, Hirsch (see [16]) stated the first fundamental result, which dealt with soluble finitelygenerated groups only. Theorem 3.1. If G is a polycyclic group then Φ(G) is nilpotent. We recall that a polycyclic group G is a soluble group with Max and so all subgroups of G are finitelygenerated. After Hirsch’s result, Itˆo asked whether or not there exist metabelian groups G with Φ(G) not nilpotent. The answer of this question is negative as P. Hall [15] showed. Example 3. If p ≡ 1 (mod q2)andC is a Pr¨ufer p-group, then C has an automor- phism α of order q2.LetA = α , B = αq and G = CA. The proper subgroups Cn of C are cyclic, characteristic and not maximal. Hence Φ(G)=CB is the unique maximal subgroup of G.Sinceαq − 1 represents a unit of End(C), we have [Cn,B]=Cn for all n and then Φ(G) is not locallynilpotent.
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