Preface and Introduction

Notation.

We use the standard (Bourbaki) notation: $\mathbb{N}=\{0,1,2,\ldots\}$; $\mathbb{Z}$ is the ring of integers; $\mathbb{Q}{}$ is the field of rational numbers; $\mathbb{R}{}$ is the field of real numbers; $\mathbb{C}{}$ is the field of complex numbers; $\mathbb{F}_{q}$ is a finite field with $q$ elements, where $q$ is a power of a prime number. In particular, $\mathbb{F}_{p}=\mathbb{Z}{}/p\mathbb{Z}{}$ for $p$ a prime number.

For integers $m$ and $n$, $m|n$ means that $m$ divides $n$, i.e., $n\in m\mathbb{Z}{}$. Throughout the notes, $p$ is a prime number, i.e., $p=2,3,5,7,11,\ldots,1000000007,\ldots$.

Given an equivalence relation, $[\ast]$ denotes the equivalence class containing $\ast$. The empty set is denoted by $\emptyset$. The cardinality of a set $S$ is denoted by $|S|$ (so $|S|$ is the number of elements in $S$ when $S$ is finite). Let $I$ and $A$ be sets; a family of elements of  $A$ indexed by $I$, denoted $(a_{i})_{i\in I}$, is a function $i\mapsto a_{i}\colon I\rightarrow A$.¹

Rings are required to have an identity element $1$, and homomorphisms of rings are required to take $1$ to $1$. An element $a$ of a ring is a unit if it has an inverse (element $b$ such that $ab=1=ba$). The identity element of a ring is required to act as $1$ on a module over the ring.

$% \begin{array} [c]{ll}% X\subset Y & X\text{ is a subset of }Y\text{ (not necessarily proper);}\\ X\overset{\df}{=}Y & X\text{ is defined to be }Y\text{, or equals }Y\text{ by definition;}\\ X\approx Y & X\text{ is isomorphic to }Y\text{;}\\ X\simeq Y & X\text{ and }Y\text{ are canonically isomorphic (or there is a given or unique isomorphism);}% \end{array}$

Prerequisites

An undergraduate “abstract algebra” course.

Computer algebra programs

GAP is an open source computer algebra program, emphasizing computational group theory. To get started with GAP, I recommend going to Alexander Hulpke’s page here, where you will find versions of GAP for both Windows and Macs and a guide “Abstract Algebra in GAP”. The Sage page here provides a front end for GAP and other programs. I also recommend N. Carter’s “Group Explorer”  here for exploring the structure of groups of small order. Earlier versions of these notes (v3.02) described how to use Maple for computations in group theory.

Acknowledgements

I thank the following for providing corrections and comments for earlier versions of these notes: V.V. Acharya; Yunghyun Ahn; Max Black; Tony Bruguier; Vigen Chaltikyan; Dustin Clausen; Benoît Claudon; Keith Conrad; Demetres Christofides; Adam Glesser; Darij Grinberg; Sylvan Jacques; Martin Klazar; Thomas Lamm; Mark Meckes; Max Menzies; Victor Petrov; Flavio Poletti; Diego Silvera; Efthymios Sofos; Dave Simpson; David Speyer; Kenneth Tam; Robert Thompson; Bhupendra Nath Tiwari; Leandro Vendramin; Michiel Vermeulen.

Also, I have benefited from the posts to mathoverflow by Richard Borcherds, Robin Chapman, Steve Dalton, Leonid Positselski, Noah Snyder, Richard Stanley, Qiaochu Yuan, and others.

A reference monnnn means question nnnn on mathoverflow.net and sxnnnn similarly refers to math.stackexchange.com.

The theory of groups of finite order may be said to date from the time of Cauchy. To him are due the first attempts at classification with a view to forming a theory from a number of isolated facts. Galois introduced into the theory the exceedingly important idea of a [normal] sub-group, and the corresponding division of groups into simple and composite. Moreover, by shewing that to every equation of finite degree there corresponds a group of finite order on which all the properties of the equation depend, Galois indicated how far reaching the applications of the theory might be, and thereby contributed greatly, if indirectly, to its subsequent developement.

Many additions were made, mainly by French mathematicians, during the middle part of the [nineteenth] century. The first connected exposition of the theory was given in the third edition of M. Serret’s “Cours d’Algèbre Supérieure,” which was published in 1866. This was followed in 1870 by M. Jordan’s “Traité des substitutions et des équations algébriques.”  The greater part of M. Jordan’s treatise is devoted to a developement of the ideas of Galois and to their application to the theory of equations.

No considerable progress in the theory, as apart from its applications, was made till the appearance in 1872 of Herr Sylow’s memoir “ Théorèmes sur les groupes de substitutions”  in the fifth volume of the Mathematische Annalen. Since the date of this memoir, but more especially in recent years, the theory has advanced continuously.

W. Burnside, Theory of Groups of Finite Order, 1897.

Galois introduced the concept of a normal subgroup in 1832, and Camille Jordan in the preface to his Traité… in 1870 flagged Galois’ distinction between groupes simples and groupes composées as the most important dichotomy in the theory of permutation groups. Moreover, in the Traité, Jordan began building a database of finite simple groups — the alternating groups of degree at least $5$ and most of the classical projective linear groups over fields of prime cardinality. Finally, in 1872, Ludwig Sylow published his famous theorems on subgroups of prime power order.

R. Solomon, Bull. Amer. Math. Soc., 2001.

Why are the finite simple groups classifiable?

It is unlikely that there is any easy reason why a classification is possible, unless someone comes up with a completely new way to classify groups. One problem, at least with the current methods of classification via centralizers of involutions, is that every simple group has to be tested to see if it leads to new simple groups containing it in the centralizer of an involution. For example, when the baby monster was discovered, it had a double cover, which was a potential centralizer of an involution in a larger simple group, which turned out to be the monster. The monster happens to have no double cover so the process stopped there, but without checking every finite simple group there seems no obvious reason why one cannot have an infinite chain of larger and larger sporadic groups, each of which has a double cover that is a centralizer of an involution in the next one. Because of this problem (among others), it was unclear until quite late in the classification whether there would be a finite or infinite number of sporadic groups.

Richard Borcherds, mo38161.