Group theory is the branch of mathematics that studies groups, which are sets of elements combined with an operation that satisfies closure, associativity, identity, and invertibility. It is widely used to analyze symmetry, transformations, and algebraic structures.
Algebraic Structures
Algebraic structures are mathematical systems made up of a set of elements and one or more operations (like addition or multiplication) that follow specific rules. Examples : Groups, Rings, fields, etc.
- Set: A collection of distinct objects or elements.
Example: A={1,2,3}.
- Semigroup : A set with an operation that is closed (result stays in the set) and associative.
Closure: Adding two natural numbers gives another natural number.
Associativity: (a + b) + c = a + (b + c).
Example: The set of natural numbers {1,2,3,… } with addition.
- Monoid: A semigroup with an identity element.
Identity: 0 is the identity because a+0=a.
Example: The set of natural numbers {0,1,2,… } with addition.
- Group: A monoid where every element has an inverse.
Inverse: For any a, there exists - a such that a+(−a)=0.
Example: The set of integers Z with addition
- Abelian Group (Commutative Group): A group where the operation is also commutative (order doesn’t matter).
Commutativity : a + b = b + a.
Example: The set of integers Z with addition.
- Ring: A set with two operations (e.g., addition and multiplication) satisfying:
- Addition forms an abelian group.
- Multiplication forms a semigroup.
- Multiplication distributes over addition.
Example: The set of integers Z with addition and multiplication: a(b + c) = ab + ac.
- Field: A ring where:
- Addition forms an abelian group.
- Multiplication (excluding 0) forms an abelian group.
Example: The set of rational numbers Q with addition and multiplication:
Every nonzero number has a multiplicative inverse, e.g., 2×1/2=1.
- Lattice: A partially ordered set where every pair of elements has a least upper bound (join) and a greatest lower bound (meet).
Example: The set of divisors of 12 ({1,2,3,4,6,12}) under divisibility.
Group
A group is a set of elements combined with a binary operation (like addition or multiplication) that satisfies four properties: closure, associativity, identity, and invertibility and it is represented as ( G, * ) where G is the set and * is the binary operation.
Example : ( Z, +) set of integers with addition forms a group.
Finite Group : A group with finite number of elements is called a finite group. where O(G) represents the order of the finite group (i.e. number of elements).
Properties of Group
- Closure: If a and b are elements of the group G, then a ⋅ b ∈G. The result of the group operation on any two elements is also in the group.
- Associativity: The group operation is associative, meaning (a ⋅ b)⋅c=a⋅(b ⋅ c) for all a, b, c ∈ G.
- Identity Element: There exists an identity element e ∈ G such that a ⋅ e = e ⋅ a=a for every a ∈ G.
- Inverse Element: For every element a ∈ G, there exists an inverse element a−1 ∈G such that a⋅a−1 = a−1⋅a = e, where e is the identity element.
- If G is a group and a, b ∈ G then (ab)-1 = b-1 a-1
- Uniqueness of Identity: The identity element in a group is unique.
- Uniqueness of Inverses: Each element in a group has a unique inverse.
- Cancellation Law:
- Left Cancellation: If a ⋅ b = a ⋅ c, then b = c.
- Right Cancellation: If b ⋅ a = c ⋅ a, then b = c.
- Group Order: The order of a group is the total number of elements in the group. For finite groups, this is a positive integer.
- Commutativity (for Abelian Groups): In an Abelian group, a ⋅ b = b ⋅ a for all a, b ∈ G. This property is specific to abelian groups.
- Power Laws (in Multiplicative Notation):
- am ⋅ an = am + n.
- (am)n = am⋅n.
- a0 = e, where e is the identity element.
Sub Group
A set H is called a subgroup of G if H ⊆ G and satisfies the group properties. i..e. ( H, * ) is also a group.
Some Important Points Related to Subgroup
- A non-empty subset H of G is a subgroup if H satisfies the closure property and contains inverse for all elements in it.
- The set {e}, containing only the identity element, and G itself are called trivial subgroup of any group G.
- The intersection of two or more subgroups of G is also a subgroup of G. i.e. if H and K are sub groups then H ∩ K is also a sub group.
- The union of two or more subgroups of G is not necessarily a subgroup of G.
- If H and K are sub groups and H ⊆ K then H ⋃ K is also a sub group.
- The order of subgroup must divide the order of the group. ( Langrange's Theorem ).
- A subgroup generated by an element a ∈ G is called a cyclic subgroup and is denoted by ⟨a⟩={an : n ∈ Z}.
Cyclic Group
A group is called a cyclic group if there exist an element of a ∈ G for which every element of a can be written as an for some integer n. And here a is called generating element/generator.
If an is the generating element for a cyclic group then am is also a generator of G if GCD(m, n) = 1
Some Important Points Related to Cyclic Group
- If a is a generator of a cyclic group then a-1 is also a generator of the cyclic group.
- Every cyclic group is an abelian group.
- For a cyclic group the number of generator is given by ϕ(n) (Euler's Phi Function ).
- Every group of prime order cyclic group hence an abelian group.
- Every subgroup of a cyclic group is also a cyclic group but the generator of the subgroup need not be the same as that of the group.
- If G is a group of even order then there exist at-least one element a ∈ G such that a-1 = a.