3.2 Groups: Definition and Examples

Groups are the foundation of abstract algebra, defining sets with special operations that follow key rules. They're like mathematical building blocks, appearing in various forms across math and science.

Understanding groups helps us see patterns in numbers, geometry, and more. We'll explore different types of groups, from simple number systems to complex mathematical structures, and learn how to identify and work with them.

Group Definition and Axioms

Defining a Group

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• A group is a set $[G](https://www.fiveableKeyTerm:g)$ together with a binary operation $*$ satisfying specific axioms
• The binary operation $*$ combines any two elements $a, b \in G$ to produce another element $a * b \in G$
• The axioms that define a group ensure the structure has certain desirable properties

Group Axioms

• Closure: For all $a, b \in G$, the result of the operation $a * b$ is also in $G$
  • The binary operation must always produce an element within the set
  • Example: The integers under addition $([Z](https://www.fiveableKeyTerm:z), +)$ are closed since the sum of any two integers is an integer
• Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$
  • The order of applying the binary operation to three elements does not matter
  • Example: For real numbers under multiplication $([R](https://www.fiveableKeyTerm:r), \cdot)$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in R$
• Identity element: There exists an element $e \in G$ such that for all $a \in G$, $a * e = e * a = a$
  • The identity element leaves any element unchanged when combined with it using the binary operation
  • Example: In the group of integers under addition $(Z, +)$, the identity element is $0$ since $a + 0 = 0 + a = a$ for all $a \in Z$
• Inverse elements: For each $a \in G$, there exists an element $b \in G$ such that $a * b = b * a = e$, where $e$ is the identity element
  • Each element in the group must have an inverse that, when combined with the element using the binary operation, produces the identity element
  • Example: In the group of nonzero real numbers under multiplication (R\{0}, \cdot), the inverse of each number $a$ is its reciprocal $a^{-1}$ since $a \cdot a^{-1} = a^{-1} \cdot a = 1$

Verifying Group Structures

Checking Group Axioms

• To verify if a set with a binary operation forms a group, check if it satisfies all four group axioms
• Closure can be verified by checking if the result of the binary operation on any two elements in the set is also an element of the set
  • Example: For the set of integers under subtraction $(Z, -)$, closure does not hold since $1 - 2 = -1$ is an integer, but $1 - (-1) = 2$ is not
• Associativity can be verified by checking if the binary operation is associative for all possible combinations of three elements in the set
  • Example: For the set of 2 × 2 matrices under matrix multiplication, associativity holds since $(AB)C = A(BC)$ for all 2 × 2 matrices $A, B, C$
• The identity element can be found by checking if there exists an element that, when combined with any other element using the binary operation, results in the same element
  • Example: In the group of invertible $n \times n$ matrices under matrix multiplication, the identity element is the $n \times n$ identity matrix $I_n$
• Inverse elements can be found by checking if, for each element in the set, there exists another element that, when combined with the original element using the binary operation, results in the identity element
  • Example: In the group of permutations of $n$ elements under composition, the inverse of each permutation is its inverse permutation, which undoes the original permutation

Subgroup Test

• A non-empty subset $H$ of a group $G$ is a subgroup if and only if for all $a, b \in H$, $a * b^{-1}$ is also in $H$
• This test simplifies the process of verifying if a subset of a group is itself a group under the same binary operation
  • Example: The set of even integers under addition $(2Z, +)$ is a subgroup of the group of integers under addition $(Z, +)$ since the difference of any two even integers is even

Examples of Groups

Integer and Real Number Groups

• The integers under addition $(Z, +)$ form a group, with $0$ as the identity element and the negative of each integer as its inverse
• The nonzero real numbers under multiplication (R\{0}, \cdot) form a group, with $1$ as the identity element and the reciprocal of each number as its inverse
  • Example: In (R\{0}, \cdot), the inverse of $2$ is $\frac{1}{2}$ since $2 \cdot \frac{1}{2} = \frac{1}{2} \cdot 2 = 1$

Matrix and Permutation Groups

• The set of $n \times n$ invertible matrices under matrix multiplication forms a group, known as the general linear group $[GL(n, R)](https://www.fiveableKeyTerm:gl(n,_r))$, with the identity matrix as the identity element and the inverse matrix of each element as its inverse
  • Example: In $GL(2, R)$, the inverse of the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, assuming $ad-bc \neq 0$
• The set of permutations of $n$ elements, denoted as $[S_n](https://www.fiveableKeyTerm:s_n)$, forms a group under the composition of permutations, with the identity permutation as the identity element and the inverse permutation of each element as its inverse
  • Example: In $S_3$, the permutation $(1 2 3)$ has the inverse permutation $(1 3 2)$ since $(1 2 3) \circ (1 3 2) = (1 3 2) \circ (1 2 3) = (1)(2)(3)$, the identity permutation

Identifying Identity and Inverses

Identity Element

• The identity element $e$ in a group is the element that, when combined with any other element $a$ using the binary operation, results in $a$: $a * e = e * a = a$
• In the group of integers under addition $(Z, +)$, the identity element is $0$
  • Example: For any integer $a$, $a + 0 = 0 + a = a$
• In the group of nonzero real numbers under multiplication (R\{0}, \cdot), the identity element is $1$
  • Example: For any nonzero real number $a$, $a \cdot 1 = 1 \cdot a = a$

Inverse Elements

• The inverse of an element $a$ in a group is the element $b$ such that $a * b = b * a = e$, where $e$ is the identity element
• In the group of integers under addition $(Z, +)$, the inverse of each integer is its negative
  • Example: The inverse of $5$ is $-5$ since $5 + (-5) = (-5) + 5 = 0$
• In the group of nonzero real numbers under multiplication (R\{0}, \cdot), the inverse of each number is its reciprocal
  • Example: The inverse of $\frac{2}{3}$ is $\frac{3}{2}$ since $\frac{2}{3} \cdot \frac{3}{2} = \frac{3}{2} \cdot \frac{2}{3} = 1$

Applications of Group Properties

Simplifying Expressions and Solving Equations

• Group properties, such as closure, associativity, identity element, and inverse elements, can be used to simplify expressions and solve equations involving group elements
• The cancellation law, which states that if $a * b = a * c$ or $b * a = c * a$, then $b = c$, can be used to solve equations in groups
  • Example: In the group of integers under addition $(Z, +)$, if $x + 5 = 3$, then by adding $-5$ to both sides, we get $x = 3 + (-5) = -2$

Proving Theorems

• The uniqueness of the identity element and inverses can be proved using the group axioms and the cancellation law
  • Example: To prove the uniqueness of the identity element, suppose $e$ and $e'$ are both identity elements. Then $e = e * e' = e'$, so the identity element is unique
• Subgroups can be identified using the subgroup test: a non-empty subset $H$ of a group $G$ is a subgroup if and only if for all $a, b \in H$, $a * b^{-1}$ is also in $H$
  • Example: The set of rotations of a square forms a subgroup of the symmetry group of the square since the composition of any two rotations is a rotation, and the inverse of a rotation is a rotation