6.1 Groups, subgroups, and group actions

Groups, subgroups, and group actions form the foundation of abstract algebra. These concepts help us understand the structure and symmetry in mathematical systems, from simple number operations to complex geometric transformations.

By studying these ideas, we gain powerful tools for analyzing and solving problems in various fields. Group theory's applications range from cryptography to quantum mechanics, making it a crucial area of study in modern mathematics and science.

Groups and their properties

Definition and axioms

• A group is a set $G$ together with a binary operation $*$ satisfying the group axioms:
  • Closure: For all $a, b \in G$, $a * b \in G$
  • Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$
  • Identity: There exists an element $e \in G$ such that $a * e = e * a = a$ for all $a \in G$
  • Inverses: For each $a \in G$, there exists an element $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$

Group properties

• The order of a group is the number of elements in the group
  • A group is finite if its order is finite, and infinite otherwise
• A group is abelian (or commutative) if $a * b = b * a$ for all $a, b \in G$
  • Examples of abelian groups include $(\mathbb{Z}, +)$, $(\mathbb{R}, +)$, and $(\mathbb{R}^*, \cdot)$
• A group is cyclic if it can be generated by a single element
  • For each positive integer $n$, there exists a unique cyclic group of order $n$ (up to isomorphism), denoted $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}_n$
• The center of a group $G$ is the set of elements that commute with every element of $G$
  • The center is always a subgroup of $G$
• The direct product of two groups $(G, *)$ and $(H, \circ)$ is the group $(G \times H, \cdot)$ where the operation $\cdot$ is defined by $(g_1, h_1) \cdot (g_2, h_2) = (g_1 * g_2, h_1 \circ h_2)$

Subgroups and their relationships

Subgroup definition and test

• A subset $H$ of a group $G$ is a subgroup if $H$ is itself a group under the same operation as $G$
  • The subgroup test states that $H$ is a subgroup of $G$ if and only if $H$ is nonempty and closed under the group operation and inverses
• The trivial subgroup $\{e\}$ and the group $G$ itself are always subgroups of $G$
  • A proper subgroup is a subgroup that is not the whole group

Cosets and Lagrange's Theorem

• If $H$ is a subgroup of $G$, then the left cosets of $H$ in $G$ are the sets of the form $aH = \{ah : h \in H\}$ for each $a \in G$
  • Similarly, the right cosets are the sets $Ha = \{ha : h \in H\}$
  • In general, left and right cosets may differ, but they coincide when $H$ is a normal subgroup
• Lagrange's Theorem states that if $G$ is a finite group and $H$ is a subgroup of $G$, then the order of $H$ divides the order of $G$
  • As a consequence, the order of any element of a finite group divides the order of the group
• The index of a subgroup $H$ in $G$, denoted $[G:H]$, is the number of left (or right) cosets of $H$ in $G$
  • Lagrange's Theorem implies that $[G:H] = |G| / |H|$

Normal subgroups

• A normal subgroup is a subgroup $N$ of $G$ such that $gNg^{-1} = N$ for all $g \in G$
  • Every subgroup of an abelian group is normal
  • The kernel of a group homomorphism is always a normal subgroup of the domain

Group actions

Definition and properties

• A group action of a group $G$ on a set $X$ is a function $\cdot : G \times X \to X$ satisfying the following properties:
  1. $e \cdot x = x$ for all $x \in X$, where $e$ is the identity element of $G$
  2. $(gh) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$
• The orbit of an element $x \in X$ under the action of $G$ is the set $\text{Orb}(x) = \{g \cdot x : g \in G\}$
  • The orbits form a partition of $X$
• The stabilizer of an element $x \in X$ is the set $\text{Stab}(x) = \{g \in G : g \cdot x = x\}$
  • The stabilizer is always a subgroup of $G$

Orbit-Stabilizer Theorem and types of actions

• The Orbit-Stabilizer Theorem states that for any $x \in X$, there is a bijection between the orbit of $x$ and the set of left cosets of $\text{Stab}(x)$ in $G$
  • As a consequence, $|\text{Orb}(x)| = [G : \text{Stab}(x)]$
• A group action is faithful if the only group element that fixes every $x \in X$ is the identity
  • Equivalently, the action is faithful if the homomorphism from $G$ to $\text{Sym}(X)$ defined by the action is injective
• A group action is transitive if for any $x, y \in X$, there exists $g \in G$ such that $g \cdot x = y$
  • Equivalently, the action is transitive if it has only one orbit

Group theorems

Subgroup and normal subgroup theorems

• Prove that the intersection of two subgroups is also a subgroup
• Prove that the center of a group is a normal subgroup
• Prove that the kernel of a group homomorphism is a normal subgroup of the domain
• Prove that if $G$ is a group and $N$ is a normal subgroup of $G$, then the quotient group $G/N$, consisting of cosets of $N$ in $G$, is a well-defined group under the operation $(aN)(bN) = (ab)N$

Isomorphism theorems and Cayley's Theorem

• Prove the First Isomorphism Theorem: If $\phi : G \to H$ is a group homomorphism, then $G / \ker(\phi) \cong \text{im}(\phi)$
• Prove Cayley's Theorem: Every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$
• Prove that if a group $G$ acts transitively on a set $X$ and $H$ is the stabilizer of a point $x \in X$, then there is a bijection between $X$ and the set of left cosets of $H$ in $G$