Groups and Subgroups - Core Definitions

A group is one of the most fundamental structures in abstract algebra, providing a framework for studying symmetry and transformations.

DefinitionGroup

A group is a set $G$ together with a binary operation $\cdot : G \times G \to G$ satisfying:

Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in G$
Identity: There exists $e \in G$ such that $e \cdot a = a \cdot e = a$ for all $a \in G$
Inverses: For each $a \in G$ , there exists $a^{-1} \in G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$

If additionally $a \cdot b = b \cdot a$ for all $a, b \in G$ , the group is called abelian or commutative.

ExampleClassical Examples of Groups

Integers under addition $(\mathbb{Z}, +)$ : The identity is 0, and the inverse of $n$ is $-n$
Non-zero rationals under multiplication $(\mathbb{Q}^*, \cdot)$ : The identity is 1, and the inverse of $q$ is $1/q$
Symmetric group $S_n$ : All permutations of $n$ elements under composition
General linear group $GL_n(\mathbb{R})$ : All invertible $n \times n$ matrices under multiplication

DefinitionSubgroup

A subset $H \subseteq G$ is a subgroup of $G$ (written $H \leq G$ ) if:

The identity $e \in H$
$H$ is closed under the group operation: if $a, b \in H$ , then $a \cdot b \in H$
$H$ is closed under inverses: if $a \in H$ , then $a^{-1} \in H$

Equivalently, $H \leq G$ if and only if $H$ is non-empty and for all $a, b \in H$ , we have $a \cdot b^{-1} \in H$ .

The subgroup criterion provides an efficient test for verifying subgroup structure. Every group $G$ has at least two subgroups: the trivial subgroup $\{e\}$ and $G$ itself. Subgroups that are neither trivial nor the whole group are called proper subgroups.

Remark

The subgroup structure of a group reveals much about its internal symmetry. For finite groups, Lagrange's theorem constrains possible subgroup orders, establishing that the order of any subgroup must divide the order of the group.

Understanding groups begins with recognizing patterns in concrete examples like symmetries of geometric objects, number systems, and matrix groups. The abstraction to axiomatic group theory allows these diverse phenomena to be studied uniformly.