1.1 Groups

Groups form the foundation of abstract algebra, capturing the essence of symmetry and transformation. They consist of a set and an operation that satisfy specific axioms, including closure, associativity, identity, and inverse elements.

From simple examples like integers under addition to complex structures like matrix groups, groups permeate mathematics. Subgroups, homomorphisms, and quotient groups provide tools to analyze and classify group structures, connecting abstract concepts to real-world applications in various fields.

Definition of groups

• Groups are fundamental algebraic structures that consist of a set and an operation satisfying certain axioms
• The study of groups is central to abstract algebra and has applications in various areas of mathematics, including noncommutative geometry
• Groups capture the notion of symmetry and transformation, which are key concepts in the study of geometric spaces

Closure under operation

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• For a set $G$ and an operation $*$, closure means that for any elements $a, b \in G$, the result of the operation $a * b$ is also an element of $G$
• Closure ensures that the operation on the set always yields an element within the same set
• Example: The integers under addition are closed, as the sum of any two integers is always an integer

Associativity of operation

• Associativity states that for any elements $a, b, c \in G$, $(a * b) * c = a * (b * c)$
• This property allows for the unambiguous grouping of elements when performing the operation
• Example: Matrix multiplication is associative, meaning $(AB)C = A(BC)$ for any matrices $A$, $B$, and $C$

Identity element

• An identity element $e \in G$ is an element that leaves any other element unchanged under the operation
• For any $a \in G$, $a * e = e * a = a$
• The identity element is unique for a given group
• Example: The number 0 is the identity element for the group of integers under addition

Inverse elements

• For each element $a \in G$, there exists an inverse element $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$
• Inverse elements allow for the "undoing" of the operation
• Example: In the group of nonzero real numbers under multiplication, the inverse of $a$ is $\frac{1}{a}$

Examples of groups

Integers under addition

• The set of integers $\mathbb{Z}$ forms a group under the operation of addition
• The operation is closed, associative, has an identity element (0), and each integer has an inverse (its negative)
• This group is an infinite abelian group, as the operation is commutative

Nonzero reals under multiplication

• The set of nonzero real numbers $\mathbb{R} \setminus \{0\}$ forms a group under multiplication
• The operation is closed, associative, has an identity element (1), and each nonzero real number has an inverse (its reciprocal)
• This group is also an infinite abelian group

Permutation groups

• A permutation group is a group whose elements are permutations of a given set, and the operation is the composition of permutations
• The set of all permutations of a finite set with $n$ elements is called the symmetric group, denoted $S_n$
• Permutation groups are important in the study of symmetries and have applications in various areas of mathematics

Matrix groups

• A matrix group is a group whose elements are matrices, and the operation is matrix multiplication
• Examples include:
  • The general linear group $GL(n, \mathbb{R})$, consisting of all invertible $n \times n$ matrices with real entries
  • The special linear group $SL(n, \mathbb{R})$, consisting of all $n \times n$ matrices with real entries and determinant 1
• Matrix groups play a crucial role in the study of linear transformations and have applications in physics and engineering

Subgroups

Definition of subgroups

• A subgroup $H$ of a group $G$ is a subset of $G$ that forms a group under the same operation as $G$
• For $H$ to be a subgroup, it must satisfy the following conditions:
  • $H$ is non-empty
  • $H$ is closed under the group operation
  • $H$ contains the identity element of $G$
  • For each $a \in H$, its inverse $a^{-1}$ is also in $H$

Trivial vs nontrivial subgroups

• The trivial subgroups of a group $G$ are the subgroups $\{e\}$ (containing only the identity element) and $G$ itself
• Any subgroup other than the trivial subgroups is called a nontrivial subgroup
• Example: The group of integers under addition has two trivial subgroups: $\{0\}$ and $\mathbb{Z}$

Cyclic subgroups

• A cyclic subgroup is a subgroup generated by a single element $a \in G$
• The cyclic subgroup generated by $a$ is denoted $\langle a \rangle$ and consists of all powers of $a$ (including negative powers if the group operation is written multiplicatively)
• Example: In the group of integers under addition, the subgroup generated by 2 is $\langle 2 \rangle = \{\ldots, -4, -2, 0, 2, 4, \ldots\}$

Lagrange's theorem

• Lagrange's theorem states that for a finite group $G$ and a subgroup $H$ of $G$, the order (number of elements) of $H$ divides the order of $G$
• Consequently, the order of any element $a \in G$ divides the order of $G$
• This theorem provides a powerful tool for determining the possible orders of subgroups and elements in a finite group

Group homomorphisms

Definition of homomorphisms

• A group homomorphism is a function $\phi: G \to H$ between two groups $G$ and $H$ that preserves the group operation
• For any $a, b \in G$, $\phi(a * b) = \phi(a) \star \phi(b)$, where $*$ and $\star$ are the group operations in $G$ and $H$, respectively
• Homomorphisms capture the notion of structure-preserving maps between groups

Kernel and image

• The kernel of a homomorphism $\phi: G \to H$ is the set of elements in $G$ that map to the identity element in $H$: $\ker(\phi) = \{a \in G \mid \phi(a) = e_H\}$
• The image of a homomorphism $\phi: G \to H$ is the set of elements in $H$ that are the result of applying $\phi$ to elements in $G$: $\operatorname{im}(\phi) = \{\phi(a) \mid a \in G\}$
• The kernel is always a normal subgroup of $G$, and the image is always a subgroup of $H$

Isomorphisms vs homomorphisms

• An isomorphism is a bijective (one-to-one and onto) homomorphism
• If there exists an isomorphism between two groups $G$ and $H$, they are said to be isomorphic, denoted $G \cong H$
• Isomorphic groups have the same structure and properties, differing only in the labeling of their elements
• Example: The group of integers under addition is isomorphic to the group of even integers under addition, via the isomorphism $\phi(a) = 2a$

Automorphism groups

• An automorphism of a group $G$ is an isomorphism from $G$ to itself
• The set of all automorphisms of $G$, denoted $\operatorname{Aut}(G)$, forms a group under the operation of function composition
• The study of automorphism groups provides insight into the symmetries and self-similarities of a group

Quotient groups

Cosets and normal subgroups

• For a subgroup $H$ of a group $G$ and an element $a \in G$, the left coset of $H$ with respect to $a$ is the set $aH = \{ah \mid h \in H\}$
• Similarly, the right coset of $H$ with respect to $a$ is the set $Ha = \{ha \mid h \in H\}$
• A subgroup $H$ is called a normal subgroup if, for every $a \in G$, $aH = Ha$
• Normal subgroups are essential for constructing quotient groups

Definition of quotient groups

• Given a group $G$ and a normal subgroup $N$, the quotient group (or factor group) $G/N$ is the set of all cosets of $N$ in $G$, with the operation defined by $(aN)(bN) = (ab)N$
• The quotient group $G/N$ "collapses" elements of $G$ that differ by an element of $N$, creating a new group structure
• Example: The quotient group of the integers under addition by the subgroup of even integers is isomorphic to the group of integers modulo 2

First isomorphism theorem

• The first isomorphism theorem states that for a group homomorphism $\phi: G \to H$, there is a natural isomorphism between the quotient group $G/\ker(\phi)$ and the image $\operatorname{im}(\phi)$
• This theorem provides a way to relate the structure of a group to its homomorphic images and quotient groups
• It is a fundamental result in group theory and has analogues in other algebraic structures, such as rings and modules

Applications of quotient groups

• Quotient groups have numerous applications in mathematics and physics, including:
  • Constructing new groups from existing ones
  • Classifying groups up to isomorphism
  • Studying the symmetries of geometric spaces and physical systems
• In noncommutative geometry, quotient groups are used to construct noncommutative spaces and to study their properties

Group actions

Definition of group actions

• A group action is a way of describing how a group $G$ acts on a set $X$
• Formally, a group action is a function $\cdot: G \times X \to X$ that satisfies the following conditions:
  • $e \cdot x = x$ for all $x \in X$, where $e$ is the identity element of $G$
  • $(gh) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$
• Group actions capture the notion of symmetry and transformation in various mathematical and physical contexts

Orbits and stabilizers

• For a group action of $G$ on $X$ and an element $x \in X$, the orbit of $x$ is the set $\operatorname{Orb}(x) = \{g \cdot x \mid g \in G\}$
• The orbits partition the set $X$ into disjoint subsets, each consisting of elements that are "equivalent" under the action of $G$
• The stabilizer of an element $x \in X$ is the subgroup $\operatorname{Stab}(x) = \{g \in G \mid g \cdot x = x\}$
• The stabilizer consists of all group elements that fix the element $x$ under the action

Conjugacy classes

• For a group $G$, the conjugacy class of an element $a \in G$ is the set $\operatorname{Cl}(a) = \{gag^{-1} \mid g \in G\}$
• Conjugacy classes partition the group into disjoint subsets, each consisting of elements that are "conjugate" to each other
• The study of conjugacy classes provides insight into the structure and properties of a group

Cayley's theorem

• Cayley's theorem states that every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$
• Specifically, the action is given by left multiplication: $g \cdot x = gx$ for all $g, x \in G$
• This theorem establishes a deep connection between abstract groups and permutation groups, allowing for the study of groups using the tools and techniques of permutation group theory

Classification of groups

Abelian vs non-abelian groups

• A group $G$ is called abelian (or commutative) if for all $a, b \in G$, $ab = ba$
• Groups that are not abelian are called non-abelian (or non-commutative)
• Abelian groups have a simpler structure and are easier to classify than non-abelian groups
• Example: The group of integers under addition is abelian, while the group of invertible $2 \times 2$ matrices under multiplication is non-abelian

Simple groups

• A group $G$ is called simple if it has no nontrivial normal subgroups
• Simple groups can be thought of as the "building blocks" of finite groups, as every finite group can be constructed from simple groups using extensions and direct products
• The classification of finite simple groups is a major achievement in group theory, completed in the early 1980s

Solvable groups

• A group $G$ is called solvable if it has a series of subgroups $G = G_0 \triangleright G_1 \triangleright \ldots \triangleright G_n = \{e\}$ such that each $G_i$ is normal in $G_{i-1}$ and the quotient groups $G_{i-1}/G_i$ are abelian
• Solvable groups have a "nice" structure and can be studied using techniques from the theory of abelian groups
• Example: All abelian groups and all finite $p$-groups (groups whose order is a power of a prime) are solvable

Sylow theorems

• The Sylow theorems are a set of powerful results that describe the structure of finite groups with respect to their subgroups of prime power order
• For a finite group $G$ and a prime $p$ dividing the order of $G$, the Sylow theorems state:
  • $G$ has a subgroup of order $p^k$ for each $k$ such that $p^k$ divides the order of $G$
  • All subgroups of order $p^k$ are conjugate to each other
  • The number of subgroups of order $p^k$ is congruent to 1 modulo $p$
• The Sylow theorems provide a way to study the structure of finite groups by examining their subgroups of prime power order