Permutation Groups

Donald S. Passman

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486310916
• Topic: Mathematics
• Subtopic: Group Theory
• Index: Mathematics

This volume by a prominent authority on permutation groups consists of lecture notes that provide a self-contained account of distinct classification theorems. A ready source of frequently quoted but usually inaccessible theorems, it is ideally suited for professional group theorists as well as students with a solid background in modern algebra.
The three-part treatment begins with an introductory chapter and advances to an economical development of the tools of basic group theory, including group extensions, transfer theorems, and group representations and characters. The final chapter features thorough discussions of the work of Zassenhaus on Frobenius elements and sharply transitive groups in addition to an exploration of Huppert's findings on solvable doubly transitive groups.

CHAPTER I

Introduction

1. The Symmetric Group

All groups considered in this text, except possibly for those given by generators and relations, are assumed to be finite.

Let A be a finite set and let Sym(A), for the moment, denote the set of one-to-one functions from A onto A, that is the set of permutations of A. If a ∈ A and f ∈ Sym(A), then we denote by af the image of the element a. If f, g ∈ Sym(A), then we define the function fg by a(fg) = (af)g. It is easy to see that Sym(A), with this composition multiplication, forms a group, the symmetric group on set A.

Let A and B be two finite sets of the same size. Then there is an obvious isomorphism of Sym(A) with Sym(B) which commutes with the given one-to-one correspondence between the two sets. Specifically, if ρ: A → B is the latter one-to-one correspondence and if f ∈ Sym( A), then ρ⁻¹f ρ ∈ Sym(B), and the map f

ρ⁻¹fρ from Sym(A) to Sym(B) is a group isomorphism. Thus it suffices to choose a fixed set of each finite size. Given integer n

1, if A = {1,2, …,n}, then we write Sym(n) for Sym(A). A permutation group is a subgroup of Sym(n). Here n is its degree.

Let f ∈ Sym(n), so that f is uniquely determined by the set of ordered pairs {(a, af) | a = 1, 2,… ,n}. We tilt these ordered pairs a quarter turn and write a above af. Moreover we place all these n expressions adjacent to each other and have

This correspondence is one-to-one provided we understand that the information contained in the symbol ( _a^a_f ) is just the relationship of the first row to the second. Thus the ordering of the columns is unimportant. For example, with n = 3, we have

Using this fact, we can obtain an easy rule of multiplication.

Given f,g ∈ Sym(n) and let

We reorder the columns of (_a^a_f ) until the top row looks like the bottom row of (_a^a_f ). Then

and

Hence, if the...