Characters of Solvable Groups

Finite solvable groups are characterized by P. Hall’s theorem: A finite group \(G\) is solvable if and only if for every prime number \(p\) the group \(G\) has a subgroup whose index in \(G\) equals the order of a \(p\)-Sylow subgroup of \(G\). Usual proofs use Frattini’s argument or the Schur-Zasenhaus theorem.

In general, if \(\pi\) is a set of prime integers, a finite group is a \(\pi\)-group if all prime divisors of its order lie in \(\pi\). The \(\pi\)-part of a positive integer \(n=p_1^{e_1}\cdots p_k^{e_k}\) is the factor \(p_{i_1}^{e_{i_1}}\cdots p_{i_r}^{e_{i_r}}\) where \(p_{i_j}\in\pi\). A Hall \(\pi\)-subgroup of \(G\) is a subgroup whose order is the \(\pi\)-part of the order of \(G\). Then, a subgroup \(H\) of \(G\) is a Hall \(\pi\)-subgroup if and only if \(H\) is a \(\pi\)-group and the \(\pi\)-part of its index \([G:H]\) is \(1\). Thus, Hall \(\pi\)-subgroups are generalizations of Sylow subgroups.

In this language, P. Hall’s theorem characterizes finite solvable groups in terms of the existence and properties of Hall \(\pi\)-subgroups. This is a beautiful theorem, which in particular shows how a global structure property of a finite group \(G\), namely the existence of Hall \(\pi\)-subgroups and the fact that \(G\) acts transitively on its Hall \(\pi\)-subgroups via conjugation, depends on some restrictions on the composition factors of a normal series for \(G\). Perhaps it is worth remarking that one implication in the proof of P. Hall’s theorem could be obtained using Burnside’s \(p^aq^b\)-theorem, whose first and most beautiful proof is one of the earliest application of character theory.

And this is how the book under review begins. The first part, chapters one to five, sets up the background on \(\pi\)-separable groups, several of their main properties, and their relations to the set \(\text{Irr}(G)\) of irreducible characters of a given solvable group \(G\). The second part, chapters six to eight, is devoted to several correspondences on character sets of related groups, some of them conjectural.

One such conjecture, of a local-to-global flavor, due to McKay and Alperin, is the assertion that for an arbitrary finite group \(G\) and any prime \(p\), if \(N\) is the normalizer of any \(p\)-Sylow subgroup of \(G\), the sets \[\{\chi\in\text{Irr}(G):p\nmid \chi(1)\} \qquad \text{ and } \qquad \{\chi\in\text{Irr}(N):p\nmid \chi(1)\}\] have the same cardinality. This conjecture is still open, but several cases have already been proved. In the book under review the McKay-Alperin conjecture is proved for \(p\)-solvable groups (Theorem 6.11, p. 184). For more on this intriguing conjecture, the recent monograph Character Theory and the McKay Conjecture by G. Navarro is the reference.

The third part of the book, chapters nine and ten, includes a treatment of the class of M-groups (groups \(G\) for which every irreducible character is monomial, that is, induced from a linear character of some subgroup of \(G\)), a class that includes the nilpotent groups. Here we find detailed expositions of some important results of E. C. Dade, for example that every solvable group is a subgroup of an M-group and his results on odd \(p\)-power degree monomial irreducible characters. In order to prove some of these theorems, the last chapter reviews the theory of symplectic modules, that is symplectic vector spaces with an action of the given group.

More than a century after its creation, the character theory of finite groups is still a powerful method for studying finite groups. In the book under review the focus is on the character theory of groups with many normal subgroups, such as solvable groups. The author has collected in book form many results that before were dispersed on the literature. The presentation is systematic and accessible. In fact, assuming that the reader has a solid graduate-level acquaintance with finite groups and some character theory on the level of the author’s well known textbook Character Theory of Finite Groups (AMS Chelsea, 2008), the book is self-contained. If your mathematical tastes lie within the realm of finite groups, the book will not disappoint you.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.