Chapter 2: Groups
- Page ID
- 10715
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Groups are a fundamental concept in abstract algebra. Understanding their definitions and properties, and studying examples, is essential to grasping algebraic structures.
- Define and understand the basic concepts of groups and their properties.
- Define and distinguish between various types of groups, such as finite and infinite groups, abelian and non-abelian groups, and cyclic and non-cyclic groups..
- Determine whether a given subset of a group is a subgroup.
- Prove basic results involving various types of groups.
- Solve basic problems involving various groups and their subgroups.
The following facts may be used without proof provided they have been covered in a prerequisite course or approved by your instructor. If an exercise specifically asks you to prove one of them, a proof is required.
Overview
Abstract algebra studies structures such as groups, rings, and fields. Proofs in a first course often use familiar facts about numbers, matrices, congruences, sets, and functions. This guide distinguishes those prerequisites from properties you must verify for a newly defined structure.
Functions
1. Associativity of composition: Given sets \(A, B, C, D\) and functions \(h \colon A \to B, g \colon B \to C\), and \(f \colon C \to D\),
\((f \circ g) \circ h = f \circ (g \circ h)\)
2, Bijectivity and inverse functions: A function \(f\colon A\to B\) has a two-sided inverse \(f^{-1}\colon B\to A\) if and only if \(f\) is bijective (injective and surjective).
Number Systems and Basic Arithmetic
The familiar arithmetic operations on \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb {C} \) satisfy the usual laws of arithmetic. In particular, addition and multiplication on these sets are associative and commutative, and multiplication distributes over addition. Moreover, every nonzero element of \(\mathbb{Q}, \mathbb{R}\), and \(\mathbb{C}\) has a multiplicative inverse in its respective set, whereas a nonzero integer need not have a multiplicative inverse in \(\mathbb{Z}\).
Matrix Algebra
Let \(K\in\{\mathbb{Q},\mathbb{R},\mathbb{C}\}\), let \(n\ge1\), and let \(A,B,C\in M_n(K)\), where \(M_n(K)\) denotes the set of \(n\times n\) matrices over \(K\). Then
1. Associativity of matrix addition and Matrix multiplication:
\(A+(B+C)=(A+B)+C\) and \(A(BC)=(AB)C.\)
2. Distributivity of matrix multiplication over addition:
\( A(B+C)=AB+AC, \) and \((A+B)C=AC+BC.\)
3. Commutativity of matrix addition:
\(A+B=B+A.\)
4. Noncommutativity of matrix multiplication:
5. In general, matrix multiplication is not commutative; that is, \(AB\neq BA. \)
6. Transpose and conjugate transpose:
\( (AB)^T=B^TA^T, \quad (AB)^*=B^*A^*. \)
Here for matrices over \( \mathbb{C},\) \(A^*=\overline{A}^{\,T}\); for matrices over \( \mathbb{Q},\) and \( \mathbb{R}\), \(A^*=A^T\).
7. Determinant multiplicativity:
\( \det(AB)=\det(A)\det(B),\quad \det(I_n)=1,\) where \(I_n\) denotes the \(n\times n\) identity matrix.
8. Invertibility criterion:
\(A\) is invertible if and only if \(\det(A)\ne0\).
9. Inverse of a product:
If \(A\) and \(B\) are invertible, then \( (AB)^{-1}=B^{-1}A^{-1}. \)