Book:John F. Humphreys/A Course in Group Theory
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John F. Humphreys: A Course in Group Theory
Published $\text {1996}$, 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 $x y x^{-1} = y x^4 x^{-1} = x y^3 \in N$.
Source work progress
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $14$: The classification of finite abelian groups: Proposition $14.2$