## Conjugacy Relation, Class Equation of Groups with Examples

For two elements x, y in a group G are said to be conjugate if gxg-1=y for some g in G. This relation is the conjugacy relation in group theory. It defines an equivalence relation, so the group G can be decomposed into different conjugacy classes. This way we will obtain the class equation of … Read more

## Normalizer of a Group: Definition, Example, Subgroup Proof

To make a subgroup normal in a group, we need to consider its normalizer. In this article, we will learn about normalizer of a subgroup with applications. Normalizer of a Subgroup Let H be a subgroup of a group G. Then the normalizer of H in G is defined as follows: N(H) = {g∈G: gHg-1=H}. … Read more

## Third Isomorphism Theorem: Statement, Proof

On this page, we will learn about the third isomorphism theorem for groups along with its statement and proof. Third Isomorphism Theorem Statement Let G be a group and H, K be its two normal subgroups such that H ≤ K. Then we have a group isomorphism (G/H)/(K/H) ≅ G/K. Third Isomorphism Theorem Proof First, … Read more

## Second Isomorphism Theorem: Statement, Proof

In this article, we will learn about the second isomorphism theorem for groups. This is also known as the diamond isomorphism theorem. The statement with its proof is provided below. Second Isomorphism Theorem Statement Let G be a group such that Then we have a group isomorphism H/(H∩K) ≅ HK/K. Second Isomorphism Theorem Proof The … Read more

## Orbit Stabilizer Theorem: Statement, Proof

The orbit-stabilizer theorem of groups says that the size of a finite group G is the multiplication of the size of the orbit of an element a (in A on which G acts) with that of the stabilizer of a. In this article, we will learn about what are orbits and stabilizers. We will also … Read more

## Quotient Group: Definition, Properties, Solved Examples

In this article, we will learn about the definition of quotient groups along with their examples, properties, and a few solved problems. Definition of Quotient Group Let H be a normal subgroup of a group G. Consider the set of all left cosets of H in G, that is, {aH: a∈G}. This set forms a … Read more

## Center of a Group: Definition, Example, Normal Subgroup

The center of a group G is a subset containing those elements of G that commute with every element of the group G. In this article, we will learn about the center of a group and show that it is a normal subgroup. Center of a Group Definition Let (G, o) be a group. Then … Read more

## Semigroup: Definition, Examples, Properties

A semigroup in mathematics is a set equipped with a binary operation that is associative. In this article, we will study the definition of semigroups together with examples, and properties. Definition of a Semigroup Let G be a non-empty set and o be an algebraic operation acting on it. Then the pair (G, o) is … Read more

## Group Theory: Definition, Examples, Properties

In Group theory, we analyze the algebraic structures of a set with a binary operation given. In this article, we will learn the definition of a group (in Abstract Algebra) with their properties, examples, and applications. Definition of a Group Let G be a set and o be a binary operation acting on it. Then … Read more

## Simple Group: Definition, Examples, Properties, Classification

A simple group is basically a group having no proper nontrivial normal subgroups. For example, A5 is a simple group. In this post, we will learn about simple groups with examples, properties, and classification. Definition of Simple Group A group is called a simple group if its only normal subgroups are the trivial subgroup and … Read more