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Group theory, in modern algebra, the study of groups, which are systems consisting of a set of elements and a binary operation that can be applied to two elements of the set, which together satisfy certain axioms. These require that the group be closed under the operation (the combination of any two elements produces another element of the group), that it obey the associative law, that it contain an identity element (which, combined with any other element, leaves the latter unchanged), and that each element have an inverse (which combines with an element to produce the identity element). If the group also satisfies the commutative law, it is called a commutative, or abelian, group.

The set of integers under addition, where the identity element is 0 and the inverse is the negative of a positive number or vice versa, is an abelian group. Groups are vital to modern algebra; their basic structure can be found in many mathematical phenomena. Groups can be found in geometry, representing phenomena such as symmetry and certain types of transformations. Group theory has applications in physics, chemistry, and computer science, and even puzzles like Rubik’s Cube can be represented using group theory.

Group Theory: Types of Cyclic Groups with Mathematical Examples | TIFR | NNHM | ISI M.MATH | Part 24

Group Theory | Subgroup | Subgroup Theorems | TIFR | NNHM | ISI M.MATH | ISI MSQE | GATE NUMERICALS

Group Theory Theory Example of order of the centre of special linear group of complex number

Group Theory Mathematical examples of Centre of Special and General Linear groups Question _Solution

Group Theory Centres of Some Important Groups, Along with a Mathematical Example

Group Theory Centre of a Group, Some results on Centre of a Group Questions _ Solutions _ Answers

Group Theory: Mathematical example on Order of elements | Part 11

Group Theory: Important points on order of elements | Part 10

Group Theory Explanation of Euler's phi function with a Mathematical Example Questions _ Solutions

Group Theory Some Important results on Cyclic groups, with Mathematical Examples Question _ Solution

Group Theory Some Important results on Cyclic Groups continued, with mathematical Examples

Group Theory Some important results on Cyclic groups with Mathematical Examples Questions _ Solution

Group Theory Theory Part 25 Types of Non-Cyclic Groups with Mathematical Examples Question_ Solution

Group Theory Definition of a Cyclic Group with Explanation about the Generator Questions _ Solutions

Group Theory Definition of Simple Group with a Mathematical Example of a Question Question_ Solution

Group Theory Important Results on Normal Sub Groups with Mathematical Examples Questions_ Solutions

Group Theory Normal Sub Groups Definition and Mathematical Example Questions _ Solutions _ Answers

Group Theory Some More Important Results on Abelian Groups Questions _ Solutions _ Answers

Group Theory Mathematical Examples on Non - Abelian Groups Questions _ Solutions _ Answers

Group Theory Abelian Group, Examples on Abelian Group, Non-Abelian Group with Math Question_Solution

Group Theory Index of Sub Groups, Abelian Group with a Mathematical Example Questions _ Solutions

Group Theory COSET and Some Properties of Cosets Questions _ Solutions _ Answers

Group Theory Some Important Points on Sub Groups Questions _ Solutions _ Answers

Group Theory Equivalence Relation and Some Remarks on Equivalence Relation Questions _ Solutions

Group Theory: Properties of Group, Definition of Ring with examples, Commutative Ring | Part 7

Group Theory Example G = {0,1,2,3} doesn't form a Group when x star y is defined as Iy xI | PART 6

Group Theory Example to prove Zn excluding 0, Is not a Group under Multiplication Modulo 4 | PART 5

Group Theory Explanation | Multiplication Modulo N, Example of Multiplication Modulo 5 | PART 4

Group Theory: Order of General Linear Group, Special Linear Group Example of Multiplication Modulo40

PART 8 Group Theory Examples of Groups over Fields, Some Important Definitions on Order of Group

Group Theory Addition Modulo 3, Proving Z3 is a group under Addition Modulo 3 | PART 3

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