# Book:John F. Humphreys/A Course in Group Theory

## John F. Humphreys: A Course in Group Theory

Published $1996$, Oxford University Press

ISBN 0-19-853459-0.

### Contents

Preface (Liverpool, January 1996)
1 Definitions and examples
2 Maps and relations on sets
3 Elementary consequences of the definitions
4 Subgroups
5 Cosets and Lagrange's Theorem
6 Error-correcting codes
7 Normal subgroups and quotient groups
8 The Homomorphism Theorem
9 Permutations
10 The Orbit-Stabiliser Theorem
11 The Sylow Theorems
12 Applications of Sylow theory
13 Direct products
14 The classification of finite abelian groups
15 The Jordan–Hölder Theorem
16 Composition factors and chief factors
17 Soluble groups
18 Examples of soluble groups
19 Semidirect products and wreath products
20 Extensions
21 Central and cyclic extensions
22 Groups with at most 31 elements
23 The projective special linear groups
24 The Mathieu groups
25 The classification of finite simple groups
25.1 The classical groups
25.1.1 The projective special linear groups
25.1.2 The unitary groups
25.1.3 The symplectic groups
25.1.4 The orthogonal groups
25.2 Groups of Lie type
A Prerequisites from number theory and linear algebra
A.1 Number theory
A.2 Linear algebra and determinants
B Groups of order < 32
C Solutions to exercises
Bibliography
Index

## Errata

### Syndrome is Zero iff Vector is Codeword

Chapter $6$: Error-correcting codes:

Hence, the element $c$ is a codeword if and only if $c = u G = u \paren {\begin{array} {c|c} I & A \end{array} } = \paren {\begin{array} {c|c} u & v \end{array} }$, where $v = u A$, and $\paren {\begin{array} {c|c} U & v \end{array} }$ denotes the concatenation of $u$ with $v$, ...

### Product with Inverse on Homomorphic Image is Group Homomorphism

Chapter $8$: The Homomorphism Theorem:

Suppose that $H$ is an abelian group and let $\vartheta: G \to H$ be a homomorphism. Define a map $\phi: G \times G \to H$ by
$\map \phi {g_1, g_2} = \map \phi {g_1} \map \phi {g_2}^{-1}$.
Prove that $\phi$ is a homomorphism.

### Cycle Decomposition of Permutation in $S_9$

Chapter $9$: Permutations:

The cycle decomposition for the permutation
$\pi = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 1 & 4 & 6 & 2 & 8 & 9 & 7 & 5 & 3 \end{pmatrix}$
is $\begin{pmatrix} 1 \end{pmatrix} \begin{pmatrix} 2 & 4 \end{pmatrix} \begin{pmatrix} 3 & 6 & 9 \end{pmatrix} \begin{pmatrix} 5 & 8 \end{pmatrix}$. It is usual to omit the cycles of length $1$, those integers fixed by $\pi$, and so $\pi$ is abbreviated to $\begin{pmatrix} 2 & 4 \end{pmatrix} \begin{pmatrix} 3 & 6 & 9 \end{pmatrix} \begin{pmatrix} 5 & 8 \end{pmatrix}$.

### Normal Subgroup of Group of Order $24$

Chapter $12$: Applications of Sylow Theory:

Thus $S_1$ and $S_2$ are both subgroups of $\map {N_G} T$, so $H = \gen {S_1, S_2}$ is a subgroup of $\map {N_T} G$ and hence $T$ is a normal subgroup of $H$.

### Groups of Order $30$

Chapter $13$: Direct products:

To show that $x$ normalises $K$, note that
$x x^5 x^{-1} = x$, and $x y x^{-1} = x y^{-1} x^{-1} = y x^{11} x^{-1} = y x^{10} \in N$.

### Groups of Order $30$: $C_3 \times D_5$

Chapter $13$: Direct products:

Now let $N$ be the subgroup of $G$ generated by $x^3$ and $y$. Note that since
$y x^3 y^{-1} = \paren {y x y^{-1} }^3 = \paren {x^4}^3 = x^{12} = x^{-3}$,
$N$ is isomorphic to the dihedral group $\map D 3$. ...

... To show that $x$ normalises $N$, note that
$x x^3 x^{-1} = x^3$; and $x y x^{-1} = y x^4 x^{-1} = x y^3 \in N$.