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# Applications of Group Theory

• Last Updated : 12 Sep, 2022

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

Note:

• 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.
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