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

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## John F. Humphreys:

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

Published $1996$, **Oxford University Press**

- ISBN 0-19-853459-0.

### Subject Matter

### 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
- 25.3 The sporadic groups

- 25.1 The classical groups

**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 $x y x^{-1} = y x^4 x^{-1} = x y^3 \in N$.