Applications of Group Theory
Group theory is the branch of mathematics that includes the study of elements in a group. Group is the fundamental concept of algebraic structure like other algebraic structures like rings and fields.
Group: A non-empty set G with * as operation, (G, *) is called a group if it follows the closure, associativity, identity, and inverse properties.
Group Theory Properties:
If * is an operation and G is a group then, properties of group theory:
- Closure: If ‘a’ and ‘b’ are elements in group G then, a*b also belongs to group G.
- Associativity: If ‘a’, ‘b’ and ‘c’ are elements in group G then, a*(b* c) = (a*b) *c
- Identity Element: For every element ‘a’ in the group there exists an identity element e such that e*a = a*e = a
- Inverse: For all element ‘a’ belongs to group G there exists a-1 such that a*a-1 = a-1*a = e
- A non-empty set S is called a semi-group if it follows closure and associativity properties.
- A non-empty set S is called a monoid if it follows closure, associativity, and identity element properties.
- A non-empty set S is called an abelian group if it is a group and follows commutative property i.e., if a and b are elements of group G then, a*b = b*a
Classes of Group Theory:
Some classes of group theory are:
- Permutation groups: Permutation groups are mathematical groups whose elements are permutations of a given set S whose group operation is a composition of permutations in G.
- Matrix groups: Matrix group is a group G of invertible matrices with matrix multiplication as the group operation over a defined field K.
- Transformation groups: Transformation group refers to the subgroup of an automorphism group. It is like a symmetry group in that it consists of all transformations that retain a specific structure.
- Abstract groups: Abstract groups are the presentation of generators and relations. The production of factor group or quotient group, G/H of a group G by normal subgroup H is an example of abstract groups.
Branches of Group Theory:
Branches of group theory include:
- Finite group theory.
- Representation of groups.
- Lie theory.
- Combinatorics and Geometric group theory.
Applications of Group Theory:
- Group theory algorithms are used to solve Rubik’s cube.
- Many laws of Physics, Chemistry use symmetry and hence, uses group theory as it is symmetric.
- Group theory may be used to investigate any object or system attribute that is invariant under change because of its symmetry.
- Group theory is also used in harmonic analysis, combinatorics, algebraic topology, algebraic number theory, algebraic geometry, and cryptography.