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Carter Subgroups of Finite Groups E ISSN 1055-1344, Siberian Advances in Mathematics, 2009, Vol. 19, No. 1, pp. 24–74. c Allerton Press, Inc., 2009. Original Russian Text c E. P. Vdovin, 2008, published in Matematicheskie Trudy, 2008, Vol. 11, No. 2, pp. 20–106. Carter Subgroups of Finite Groups E. P. Vdovin1* 1Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia Received January 14, 2008 Abstract—It is proven that the Carter subgroups of a finite group are conjugate. A complete classification of the Carter subgroups in finite almost simple groups is also obtained. DOI: 10.3103/S1055134409010039 Key words: Carter subgroup, finite simple group, group of Lie type, linear algebraic group, semilinear group of Lie type, semilinear algebraic group, conjugated powers of an element Contents 1 INTRODUCTION 25 1 Generalcharacteristicoftheresults. ............. 25 2 NotationandresultsfromGrouptheory . .......... 26 3 Linearalgebraicgroups............................. ........ 27 4 Structure of finitegroupsofLietype ............................. 29 5 Knownresults ...................................... .... 32 2 CONJUGACY CRITERION FOR CARTER SUBGROUPS 33 1 Mainresults....................................... ..... 33 2 Preliminaryresults................................ ........ 34 3 ProofofTheorem2.1.4 ............................... ...... 35 4 SomepropertiesofCartersubgroups. .......... 35 3 CONJUGACY IN SIMPLE GROUPS 37 1 Briefreviewoftheresults........................... ......... 37 2 Preliminaryresults................................ ........ 38 3 Almost simple groups which are not minimal counter examples............. 45 4 SEMILINEAR GROUPS OF LIE TYPE 48 1 Basicdefinitions ......................................... 48 2 Translationofbasicresults . .. .. .. .......... 49 3 Cartersubgroupsofspecialtype. .......... 53 5 CARTER SUBGROUPS OF SEMILINEAR GROUPS 57 1 Briefreviewoftheresults........................... ......... 57 2 Cartersubgroupsofsymplecticgroups. ........... 57 3 Groupswithtrialityautomorphism . .......... 59 4 Classificationtheorem...................................... 60 5 Carter subgroups of orderdivisibleby characteristic . ................. 61 6 Carter subgroups of order not divisible by characteristic ................. 64 *E-mail: [email protected] 24 CARTERSUBGROUPSOFFINITEGROUPS 25 7 Cartersubgroupsof finitegroupsareconjugate. 66 6 EXISTENCE CRITERION 66 1 Briefreviewoftheresults........................... ......... 66 2 Criterion......................................... ..... 67 3 Example........................................... ... 69 4 ClassificationofCartersubgroups ............................ 69 REFERENCES 72 1. INTRODUCTION The present paper is a slightly shorten version of the doctoral thesis “Carter subgroups of finite groups”. The results of the thesis were published in [33, 39, 42–45]. 1.1. General characteristic of the results. We recall that a subgroup of a finite group is called a Carter subgroup if it is nilpotent and self-normalizing. It is well known that any finite solvable group contains exactly one conjugacy class of Carter subgroups (see [3]). If a group is not assumed to be finite then Carter subgroups can be even nonisomorphic. Indeed, if N1 and N2 are two nonisomorphic nilpotent groups then they are Carter subgroups in their free product. On the other hand, a finite nonsolvable group may fail to contain Carter subgroups, the minimal counter example is the alternating group of degree 5. Although there is not a single example of a finite group containing nonconjugate Carter subgroups, the following problem due to R. Carter is known. Problem 1.1.1. (Conjugacy Problem) Are the Carter subgroups of a finite group conjugate? This problem for several classes of finite groups close to simple was studied by many authors. For example, L. Di Martino and M. C. Tamburini classified the Carter subgroups in symmetric and alternating groups (see [14]) and also in every group G such that SLn(q) ≤ G ≤ GLn(q) (see [16]). The case G = GLn(q) was considered by the same authors earlier in [15] and, independently, by fi N. A. Vavilov in [41]. For symplectic groups Sp2n(q), general unitary groups GUn(q), and nally, for ± fi general orthogonal groups GOn (q) with odd q, the classi cation of the Carter subgroups was obtained by L. Di Martino, A. E. Zalessky, and M. C. Tamburini (see [17]). For some sporadic simple groups, Carter subgroups were found in [12]. In the nonsolvable groups mentioned above, Carter subgroups coinside with the normalizers of Sylow 2-subgroups, and hence, are conjugate. A finite group G is called a minimal counter example to Conjugacy Problem or a minimal counter example for brevity if G contains nonconjugate Carter subgroups, but in every group H, with |H| < |G|, the Carter subgroups are conjugate. In [11], F. Dalla Volta, A. Lucchini, and M. C. Tamburini proved that a minimal counter example must be almost simple. This result allows to use the classification of finite simple groups to solve Conjugacy Problem. Note that the using of the above-mentioned result by F. Dalla Volta, A. Lucchini, and M. C. Tam- burini for the classification of Carter subgroups in almost simple groups essentially depends on the classification of finite simple groups. Indeed, in order to use the inductive hypothesis that the Carter subgroups in every proper subgroup of a minimal counter example are conjugate, one needs to know that all almost simple groups of order less than the order of a minimal counter example are found. To avoid using the classification of finite simple groups, we strengthen the result from [11] proving that if Carter subgroups are conjugate in the group of induced automorphisms of every non-Abelian composition factor then they are conjugate in the group. For inductive description of the Carter subgroups in almost simple groups, one needs to know ho- momorphic images of Carter subgroups and intersections of Carter subgroups with normal subgroups, i. e., to answer the following questions: SIBERIAN ADVANCESINMATHEMATICS Vol.19 No.1 2009 26 VDOVIN Problem 1.1.2. Is a homomorphic image of a Carter subgroup again a Carter subgroup? Problem 1.1.3. Is the intersection of a Carter subgroup with a normal subgroup again a Carter subgroup (of the normal subgroup)? The first problem is closely connected with Conjugacy Problem. Namely, if Conjugacy Problem has an affirmative answer then the first problem also has an affirmative answer. So we will solve both of these problems by considering Carter subgroups in almost simple groups. It is easy to see that the second problem has a negative answer. Indeed, consider a solvable group Sym3 and its normal subgroup of index 2, i. e., the alternating group Alt3. Then a Carter subgroup of Sym3 is a Sylow 2-subgroup, while a Carter subgroup of Alt3 is a Sylow 3-subgroup. Thus, in the paper, some properties of Carter subgroups in a group and some of its normal subgroups are studied. The present paper is divided into six Sections including Introduction. In Introduction, we give general results of the paper and some necessary definitions and results as well. In Section 2, we prove that Carter subgroups of a finite group are conjugate if they are conjugate in the group of induced automorphisms of every its non-Abelian composition factor, thereby we strengthen the results by F. Dalla Volta, A. Lucchini, and M. C. Tamburini. In Section 2, we also study some properties of Carter subgroups. In Section 3, we consider the problem of conjugacy for elements of prime order in finite groups of Lie type. At the end of Section 3, using the results on conjugacy, we obtain the classification of Carter subgroups in a broad class of almost simple groups. In Section 4, we introduce the notion of semilinear groups of Lie type and the corresponding semilinear algebraic groups and transfer the results to the normalizers of p-subgroups and to the centralizers of semisimple elements in groups of Lie type. We also obtain some additional results on the conjugacy of elements of prime order in these groups. In Section 5, we complete the classification of Carter subgroups in almost simple groups and prove that the Carter subgroups of almost simple groups are conjugate. As a corollary, we obtain an affirmative answer to Conjugacy Problem and prove that a homomorphic image of a Carter subgroup is a Carter subgroup. In Section 6, we study the problem of existence of a Carter subgroup in a finite group, give a criterion of the existence and construct an example showing that the property of containing a Carter subgroup is not preserved under the extensions. Moreover, in the last Subsection of this Section, we give tables with the classification of Carter subgroups in almost simple groups. 1.2. Notation and results from Group theory. We will use the standard notation. If G is a group then H ≤ G and H E G mean that H is a subgroup and a normal subgroup of G respectively. By |G : H| we denote the index of H in G, NG(H) is the normalizer of H in G. If H is normal in G then by G/H we denote the factor group of G by H. If M is a subset of G then hMi denotes the subgroup generated by M, and |M| denotes the cardinality of M (or the order of an element if there is an element instead of a set). By CG(M) we denote the centralizer of M in G, and by Z(G) we denote the center of G. The conjugation of x by an element y in G is written as xy = y−1xy (yx = yxy−1), and by [x,y]= x−1xy we denote the commutator of x,y. The symbol [A, B] means the mutual commutant of subgroups A and B of G. For groups A and B, the expressions A × B, A ◦ B, and A ⋌ B (or B ⋋ A) mean direct, central, and semidirect products of A and B, with B normal, respectively. If S ≤ Symn and G is a group then G ≀ S denoted the permutation wreath product. If A and B are subgroups of G such that A E B, then the factor group B/A is called a section of G. The Fitting subgroup of G is denoted by F (G), the generalized Fitting subgroup is denoted by F ∗(G). The set of Sylow p-subgroups of a finite group G will be denoted by Sylp(G). If ϕ is a homomorphism ϕ ϕ of G and g is an element of G then G and g are the images of G and g under ϕ respectively.
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