Finite groups. (English) Zbl 0185.05701

Harper’s Series in Modern Mathematics. New York-Evanston-London: Harper & Row Publishers. xvi, 527 p. (1968).
Let us take a brief review on the present stage of the theory of groups of finite order, focusing, in particular, on simple groups. Definitely we have to begin with the achievement of W. Feit and J. G. Thompson on the solvability of groups of odd order [Pac. J. Math. 13, 775–1029 (1963; Zbl 0124.26402)]. This enables us to concentrate ourselves (at least to some extents) to the prime number 2, namely to the structure and behavior of 2-subgroups in simple groups. In fact, the author and J. H. Walter have successfully classified all the simple groups whose Sylow 2-subgroups are dihedral [J. Algebra 2, 85–151 (1965); 2, 218–270 (1965; Zbl 0192.11902); 3, 334–393 (1965; Zbl 0192.12001)].
Next we have to refer to the work of C. Chevalley [TĂ´hoku Math. J., II. Ser. 7, 14–66 (1955; Zbl 0066.01503)] which has completed the work of Dickson, giving a new uniform method to construct simple groups of the so-called Lie type including those corresponding to exceptional simple Lie algebras, and which has been supplemented by R. Steinberg [Pac. J. Math. 9, 875–891 (1959; Zbl 0092.02505)]. We have experienced a surprise when M. Suzuki has found, with an entirely different method, simple groups of Suzuki type [Proc. Natl. Acad. Sci. USA 46, 868–870 (1960; Zbl 0093.02301)]. Fortunately R. Ree has verified, together with a finding of simple groups of two Ree types, that they all belong to the family of simple groups of Lie type [Am. J. Math. 83, 401–420 (1961; Zbl 0104.24704); 83, 432–462 (1961; Zbl 0104.24705)]. Thirdly we have to mention that Thompson since then has succeeded in classifying all the simple groups whose proper subgroups are all solvable; they are \(L_2(q)\), \(Sz(q)\), \(A_7\), \(L_3(3)\), \(U_3(3)\) and \(M_{11}\). Thus we may have, if we can be optimistic enough, a rather rosy opinion concerning the classification problem of all simple groups of finite order, despite the threatening of “sporadic” arrival of new comers, which has begun (if we can ignore Mathieu groups) with two Janko groups [Z. Janko, Proc. Natl. Acad. Sci. USA 53, 657–658 (1965; Zbl 0142.25903)].
Now reading through this book the reviewer has got a strong impression that the author, being optimist in the above sense (and the reviewer wishes that he is ultimately right), intends to recruit new warriors to this fascinating battle field of classifying all simple groups of finite order through the presentation of this book. Although there have appeared many books on groups of finite order recently, this is the first book ever written with clear attempts going into the classification problem.
The author begins with preliminaries (Chapter 1) [Theorem 2.2 needs an obvious amendment) and with some basic topics (Chapter 2) which contains a rather thorough description of the two-dimensional linear and projective groups (§8). Chapter 3 is an introduction to the theory of (ordinary) representations of groups of finite order. But in §8 the author introduces the notion of \(p\)-stable representations; let \(p\) be an odd prime and \(\mathfrak G\) a group with no nontrivial normal \(p\)-subgroups. A faithful representation \(\Phi\) of \(\mathfrak G\) on a vector space \(\mathfrak B\) over \(\mathrm{GF}(p^n)\) is called \(p\)-stable if no \(p\)-element of \(\mathfrak G\Phi\) has a quadratic minimal polynomial on \(\mathfrak B\). Further \(\mathfrak G\) is called \(p\)-stable if all such faithful representations of \(\mathfrak G\) are \(p\)-stable. The basic result about this is
Theorem 8.3: If \(\mathfrak G\) is not \(p\)-stable, then \(\mathfrak G\) involves \(\mathrm{SL}(2,p)\). To prove this the author uses a theorem due to Baer (Theorem 8.2): Let \(\mathfrak K\) be a conjugate class of \(p\)-elements of a finite group \(\mathfrak X\). If every pair of elements of \(\mathfrak K\) generates a \(p\)-subgroup, then \(\mathfrak K\) lies in a normal \(p\)-subgroup of \(\mathfrak X\).
Chapter 4 is a standard treatment of the character theory, which includes many new results on induced and exceptional characters mainly due to Brauer, Suzuki and Feit (the proof of Theorem 2.3 is a little obscure).
Chapter 5 is devoted to groups of prime power order. It features in characteristic subgroups and extra-special groups (a \(p\)-group \(\mathfrak P\) is called to be extra-special, if \(\mathfrak P' = Z(\mathfrak P)= \Phi(\mathfrak P)\) has order \(p)\). This chapter also contains a definition of associated Lie rings.
Chapter 6 contains classical results on solvable and \(\pi\)-solvable groups. But in §5 the author proves that (Theorem 5.1) if \(\mathfrak G\) is a \(p\)-solvable group with no nontrivial normal \(p\)-subgroups and if \(p= 5\), then \(\mathfrak G\) is \(p\)-stable.
Chapter 7 treats the transfer theory.
It seems to the reviewer that Chapter 8 is the core of the book. First the author gives the definition of \(p\)-constrainedness and \(p\)-Stability (the reviewer uses the capital S to avoid the confusion with the earlier \(p\)-stability defined in Chapter 3. Let \(\mathfrak X\) be a group. Then \(O_{p'}(\mathfrak X)\) denotes the largest normal subgroup of \(\mathfrak X\) with order prime to \(p\), and \(O_{p',p}(\mathfrak X)/O_{p'}(\mathfrak X)\) denotes the largest normal \(p\)-subgroup of \(\mathfrak X/O_{p'}(\mathfrak X)\). Now a group \(\mathfrak G\) is said to be \(p\)-constrained, if for every Sylow \(p\)-subgroup \(\mathfrak G\) of \(O_{p',p}(\mathfrak G)\) the centralizer of \(\mathfrak P\) in \(\mathfrak G\), \(C_{\mathfrak G}\mathfrak P\) is contained in \(O_{p',p}(\mathfrak G)\).
(The definition of \(p\)-Stability needs an obvious adjustment, since otherwise every group becomes \(p\)-Stable. The following adjustment has been suggested by Paul Fong to the reviewer.)
Let \(\mathfrak G\) be a group such that \(O_p(\mathfrak G)\ne\mathfrak G\) and \(\mathfrak P\) be a Sylow \(p\)-subgroup of \(O_{p',p}(\mathfrak G)\). Then \(\mathfrak G\) is said to be \(p\)-Stable, if for every \(p\)-subgroup \(\mathfrak A\) of the normalizer of \(\mathfrak P\) in \(\mathfrak G\), \(N_{\mathfrak G}(\mathfrak P)\), such that \([\mathfrak P, \mathfrak A, \mathfrak A = \mathfrak G\) holds \[ \mathfrak A C_{\mathfrak G}(\mathfrak P)/(C_{\mathfrak G}(\mathfrak P)\subseteq O_p(N_{\mathfrak G}(\mathfrak P)/C_{\mathfrak G}(\mathfrak P)). \]
Now among the principal theorems proved in Chapter 8 we quote a few:
Thompson Replacement Theorem (Theorem 2.5) Let \(\mathfrak P\) be a Sylow \(p\)-subgroup of a group \(\mathfrak G\). Let \(A(\mathfrak P)\) denote the set of all abelian subgroups of \(\mathfrak P\) of maximal order. Let \(\mathfrak A\in A(\mathfrak P)\). Let \(\mathfrak B\) be an abelian subgroup of \(\mathfrak P\) such that \(\mathfrak A\subseteq N(\mathfrak B)\) but \(\mathfrak B \not\subseteq N(\mathfrak A)\). Then there exists an \(\mathfrak A^*\in A(\mathfrak P)\) such that (i) \(\mathfrak A\cap\mathfrak B\ne \mathfrak A^*\cap\mathfrak B\) and (ii) \(\mathfrak A^*\in N(\mathfrak A)\).
Theorem 2.11 (due to Glauberman). Let \(\mathfrak G\) be a group such that \(O_p(\mathfrak G) \ne \mathfrak G\), where \(p\) is an odd prime. Assume that \(\mathfrak G\) is \(p\)-constrained and \(p\)-Stable. Let \(\mathfrak P\) be a Sylow \(p\)-subgroup of \(\mathfrak G\). Then \(\mathfrak G = O_{p'}(\mathfrak G)N_{\mathfrak G} (Z(J(\mathfrak P)))\), where \(J(\mathfrak P)=\langle \mathfrak A; \mathfrak A\in A(\mathfrak P)\rangle\) (the Thompson subgroup of \(\mathfrak P)\).
Theorem 3.1 (due to Glauberman and Thompson). Let \(p\) be odd. If \(N_{\mathfrak G} (Z(J(\mathfrak P)))\) is \(p\)-nilpotent, then \(\mathfrak G\) itself is \(p\)-nilpotent.
Thompson Transitivity Theorem (Theorem 5.4). Let \(\mathfrak G\) be a group such that \(N_{\mathfrak G}(\mathfrak X)\) is \(p\)-constrained for every non-trivial \(p\)-subgroup \(\mathfrak X\) of \(\mathfrak G\). Let \(\mathrm{SCN}_3(\mathfrak P)\) denote the set of all self-centralizing subgroups \(\mathfrak Y\) of \(\mathfrak G\) such that \(\vert\mathfrak Y/\Phi(\mathfrak Y)\vert \ge p^3\). Let \(\mathfrak A\in\mathrm{SCN}_3(\mathfrak P)\). Let \(q\ne p\) be a prime and let \(\Omega\) be the set of all maximal \(\mathfrak A\)-invariant \(q\)-subgroups of \(\mathfrak G\). Then \(C_{\mathfrak G}(\mathfrak A)/\Omega\) is transitive (of course, the transformation is the conjugation).
The Maximal Subgroup Theorem (Theorem 6.3). Let \(\mathfrak P\) be a Sylow \(p\)-subgroup of a group \(\mathfrak G\) such that \(\mathrm{SCN}_3(\mathfrak P)\) is not empty, where \(p\) is odd. Let \(A_1(\mathfrak P)\) be the set of all subgroups of \(\mathfrak P\) which contains at least one element of \(\mathrm{SCN}_3(\mathfrak P)\). For \(i>1\) define inductively, \(A_i(\mathfrak P)\) is the set of all subgroups \(\mathfrak Q\) of \(\mathfrak P\) such that \(\mathfrak Q\) contains a subgroup \(\mathfrak R\) of type \((p,p)\) with \(C_{\mathfrak P}(X) \in A_{i -1}(\mathfrak P)\) for every element \(X\ne E\) of \(\mathfrak R\). Finally \(N^*(\mathfrak P)\) is the set of all subgroups \(\mathfrak H\) of \(\mathfrak G\) of the form \(\mathfrak H = N_{\mathfrak G} (\mathfrak Q)\), \(\mathfrak G\ne \mathfrak Q\subseteq \mathfrak P\) such that \(\mathfrak H\) contains at least one member of \(A_i(\mathfrak P)\) for some \(i\). Now the assertion says that \(N^*(\mathfrak P)\) contains the largest element if the following two conditions are satisfied: (i) Every element of \(N^*(\mathfrak P)\) is \(p\)-constrained and \(p\)-Stable. (ii) There exists a non-trivial subgroup \(\mathfrak Q\) of \(\mathfrak P\) such that if \(\mathfrak X\) is a \(p'\)-subgroup of \(\mathfrak G\) such that \(N^*(\mathfrak X)\supseteq\mathfrak P\) then \([\mathfrak X, \mathfrak Q] = \mathfrak G\).
Chapter 9 contains classical results concerning involutions together with a so-called group order formula.
Chapter 10 is given to proofs of the nilpotency theorem of Thompson concerning groups with fixed-point-free automorphism of prime order (Theorem 2.1) and its generalizations.
Chapter 11 is devoted to the so-called Hall-Higman theorem which may be regarded as a part of the theory of modular representations.
Chapter 12 is given to the proof of a theorem by Brauer and Suzuki (Theorem 1.1): If a Sylow 2-subgroup \(\mathfrak S\) of a group \(\mathfrak G\) is a generalized quaternion group of order at least 16, then \(\mathfrak G\) is not simple (the case where \(\mathfrak S\) has order 8 is omitted because of the lack of a non-modular proof).
Chapter 13 discusses Zassenhaus groups, namely doubly transitive groups in which only the identity fixes three letters. The fundamental theorem of Feit is proved (Theorem 2.1): If a Zassenhaus group \(\mathfrak G\) is simple and of degree \(n+1\), then \(n\) is a prime power. Then a certain class of Zassenhaus groups is identified with the class of \(L_2(n)\) with \(n>3\).
Chapter 14 treats groups in which centralizers (of non-identity elements) are nilpotent. Such a group \(\mathfrak G\) is called a CN-group. Two principal theorems are proved:
Theorem 3.1 (due to Feit, M. Hall, Thompson). If a CN-group has an odd order, then \(\mathfrak G\) is solvable.
Theorem 4.1 (Suzuki). If a CN-group \(\mathfrak G\) is a simple group of composite order with an abelian Sylow 2-subgroup \(\mathfrak S\), then \(\mathfrak G\) is isomorphic with \(L_2(\vert\mathfrak S)\vert\).
Chapter 15 is devoted to the classification problem of groups with self-centralizing Sylow 2-subgroups of order 4 (due to the author and Walter). If such a group \(\mathfrak G\) is simple, then \(\mathfrak G\) is isomorphic with \(L_2(q)\), \(q\equiv 3,5\pmod 8\), \(q>3\) (Theorem 2.1).
Chapter 16 is very interesting. The author gives his own analyses on recent major works on the classification problem of simple groups of finite order. The works taken are:
(1) The Feit-Thompson theorem on the solvability of groups of odd order.
(2) The theorem by the author and Walter on groups with dihedral Sylow 2-subgroups.
(3) The theorem by Suzuki on C-groups, where a group \(\mathfrak G\) is called a C-group if for any involution \(x\) of \(\mathfrak G\) the centralizer of \(x\) in \(\mathfrak G\) possesses the normal Sylow 2-subgroup.
(4) The theorem by Thompson on N-groups, where a group \(\mathfrak G\) is called an N-group if the normalizer of every non-trivial solvable subgroup of \(\mathfrak G\) is solvable.
(5) The theorem by Walter on groups with abelian Sylow 2-subgroups (including the try so on Janko groups).
The chapter ends with the reference to not so complete (comparing with (1) – (6)) classification theorems together with some open problems.
The final chapter (Chapter 17) contains the list of all known simple groups (at the time of the publication of this book), and moreover, frank and inspiring discussions by the author on future classification problems of simple groups.
Reviewer: Noboru Ito


20-02 Research exposition (monographs, survey articles) pertaining to group theory
20D05 Finite simple groups and their classification
20Dxx Abstract finite groups
20E32 Simple groups
20Exx Structure and classification of infinite or finite groups