Characters of solvable groups

384 Pages·2018·2.041 MB·English

by  Isaacs, I. Martin

GRADUATE STUDIES 189
IN MATHEMATICS
(cid:39)(cid:76)(cid:69)(cid:86)(cid:69)(cid:71)(cid:88)(cid:73)(cid:86)(cid:87)(cid:3)(cid:83)(cid:74)(cid:3)
Solvable Groups
I. Martin Isaacs
Characters of
Solvable Groups

GRADUATE STUDIES 189
IN MATHEMATICS
Characters of
Solvable Groups
I. Martin Isaacs
EDITORIAL COMMITTEE
Dan Abramovich
Daniel S. Freed (Chair)
Gigliola Staffilani
Jeff A. Viaclovsky
2010 Mathematics Subject Classification. Primary 20C15;
Secondary 20C20, 20D10, 20D35, 20F16, 20F14.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-189
Library of Congress Cataloging-in-Publication Data
Names: Isaacs,I.Martin,1940–author.
Title: Charactersofsolvablegroups/I.MartinIsaacs.
Othertitles: Solvablegroups
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2018]|Series: Gradu-
atestudiesinmathematics;volume189|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2017047819|ISBN9781470434854(alk. paper)
Subjects: LCSH: Solvable groups. | Group theory. | AMS: Group theory and generalizations
– Representation theory of groups – Ordinary representations and characters. msc | Group
theory and generalizations – Representation theory of groups – Modular representations and
characters. msc|Grouptheoryandgeneralizations–Abstractfinitegroups–Solvablegroups,
theory of formations, Schunck classes, Fitting classes, π-length, ranks. msc | Group theory
andgeneralizations–Abstractfinitegroups–Subnormalsubgroups. msc|Grouptheoryand
generalizations – Special aspects of infinite or finite groups – Solvable groups, supersolvable
groups. msc|Grouptheoryandgeneralizations–Specialaspectsofinfiniteorfinitegroups–
Derivedseries,centralseries,andgeneralizations. msc
Classification: LCCQA177.I83252018|DDC512/.2—dc23
LCrecordavailableathttps://lccn.loc.gov/2017047819
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Contents
Preface ix
Part 1. π-Theory
Chapter 1. π-Separable Groups and Character Theory Background 3
Chapter 2. π-Special Characters 25
Chapter 3. Partial Characters 67
Chapter 4. The Nucleus and B -Characters 95
π
Chapter 5. B (G) and I (G) 131
π π
Part 2. Character Counts and Correspondences
Chapter 6. The Okuyama–Wajima Argument 173
Chapter 7. Fully Ramified Abelian Sections 209
Chapter 8. Fully Ramified Sections and Character Correspondences 245
Part 3. M-Groups
Chapter 9. M-Groups and Monomial Characters 287
Chapter 10. Symplectic Modules and Character Theory 311
Bibliography 359
Index 363
vii

Preface
When character theory was developed at the end of the 19th century and in
the early years of the 20th, it was viewed primarily as a powerful technique
for proving theorems about finite groups. For example, Burnside’s paqb-
theorem and Frobenius’s theorem on the existence of what we now call
Frobenius kernels were early triumphs of the use of character theory as a
tool for proving group-theory theorems. Indeed, characters continued to
play an essential role in proving theorems about groups for another 50 years
or so, culminating in the Feit–Thompson odd-order theorem.
Another thread in the development of character theory is the establish-
ment of connections between the characters and the subgroup structure of a
group. Examples of this are Brauer’s theorem on induced characters, which
asserts that every character of G is a Z-linear combination of characters
induced from certain nilpotent subgroups, and also the Itˆo–Michler theo-
rem, which gives a necessary and sufficient condition in terms of irreducible
character degrees for G to have a normal abelian Sylow p-subgroup.
Researchontheconnectionsbetweencharactersandsubgroupscontinues
unabated, and there are still a number of intriguing open questions in this
area. For example, consider the still-unresolved McKay conjecture, which
posits that for each finite group G and each prime number p, the numbers
of irreducible characters with degrees not divisible by p of G and of the
normalizer of a Sylow p-subgroup of G are equal.
Within the last half century or so, there has been considerable atten-
tion paid to the character theory of groups such as solvable and p-solvable
groups that have an abundance of normal subgroups. One reason for this is
that groups of this type often serve to motivate and to “test” more general
questions. For example, the Itoˆ–Michler theorem was initially proved (by
ix