ankit prajapati

Group Theory – Short Description

Equivalence Relation and Partition:

An equivalence relation is a relation that satisfies the properties of reflexivity, symmetry, and transitivity. It divides a set into disjoint equivalence classes, which together form a partition of the set.

Congruence Modulo (n):

Two integers (a) and (b) are said to be congruent modulo (n) if their difference ((a-b)) is divisible by (n). It is denoted by (a \equiv b \pmod{n}).

Group:

A group is an algebraic structure consisting of a set and a binary operation that satisfies the closure, associative, identity, and inverse properties.

Simple Properties of a Group:

The identity element is unique, every element has a unique inverse, and the cancellation law holds in a group.

Subgroup:

A subgroup is a non-empty subset of a group that is itself a group under the same binary operation.

Generator of a Group:

A generator is an element from which every element of the group can be obtained by repeated application of the group operation.

Cyclic Group:

A cyclic group is a group that can be generated by a single element. Every element of the group is obtained from that generator.