## 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

## Normal Subgroup: Definition, Examples, Properties, Theorems

A normal subgroup H of a group G is a subgroup of G that is invariant under conjugation by members of the group. In other words, every left coset and right coset corresponding to an element g are the same, that is, gH=Hg. Normal subgroups have many applications. In this post, we will learn about normal subgroups with … Read more

## Left Cosets and Right Cosets: Definition, Examples, Properties, Theorems

Cosets are mainly used to decompose a group G into equal-sized disjoint subsets of G. It plays an important role to study many things in Group Theory; for example, normal group, Lagrange’s theorem on finite groups, etc. In this post, we will learn about cosets, their classification with examples, and their properties with related theorems. … Read more

## Order of a Permutation: Definition, Examples, How to Find

A bijective mapping on a finite set S is called a permutation on S. In this post, we will discuss the order of a permutation, how to find the order of a permutation with examples, and related theorems. Definition of Order of a Permutation Order of permutation:- The order of a permutation σ on a … Read more

## Group Isomorphism: Definition, Properties, Examples

An isomorphism of groups is a special kind of group homomorphisms. It preserves every structure of groups. In this article, we will learn about isomorphism between groups, related theorems, and applications. Definition of Isomorphism A map Φ: (G, 0) → (G′, *) between two groups is called an isomorphism if the following conditions are satisfied: Φ … Read more

## Group Homomorphism: Definition, Examples, Properties

A group homomorphism is a map between two groups that preserves the algebraic structure of both groups. In this section, we will learn about group homomorphism, related theorems, and their applications. Definition of Group Homomorphism A map Φ: G → G′ between two groups  (G, 0) and (G′, *) is called a group homomorphism if the … Read more

## Kernel of a Homomorphism

The kernel of a group homomorphism is an interesting subgroup of the domain group. This subgroup (kernel) determines whether it is an injective homomorphism or not. In this article, we will learn about the kernel of group homomorphisms. What is a kernel (Algebra)? Definition of a Kernel of a Homomorphism Let Φ: (G, 0) → … Read more

## Group of Prime Order

If a group has order p and p is a prime, then we call that group to be a group of prime order. A group of prime order has a nice description, and they can be characterized as follows: A group of prime order is cyclic. A group of prime order p is isomorphic to … Read more

## Abelian Group: Definition, Properties, Examples

Abelian groups are special types of groups in which commutativity holds. In other words, the binary operation on such groups is commutative. Abelian groups are named after mathematician Niels Henrik Abel. In this article, we will discuss abelian groups with their properties. What is an Abelian Group? A group (G, o)  is called an abelian … Read more

## First Isomorphism Theorem: Proof and Application

The first isomorphism theorem for groups proves that every homomorphic image of a group is actually a quotient group. This theorem is also known as the fundamental theorem of homomorphism. In this article, we will learn about the first isomorphism theorem for groups and the theorem is given below. First isomorphism theorem of groups: Let … Read more