## Group of Order 4 is Abelian: Proof

A group of order 4 is always abelian or commutative. That is, ab = ba for all a, b in G where |G| =4. In this article, we will prove that each group of order 4 is abelian. What are Abelian Groups? A pair (G, o) is called an abelian group if G is closed … Read more

## Two Cyclic Groups of Same Order are Isomorphic

Cyclic groups of same order are isomorphic. So there is only one cyclic group of order n, up to isomorphism. In this article, we will prove that two cyclic groups of same order are isomorphic. Proof Let us consider two cyclic groups G and G’ of same order generated by a and b respectively. Thus, … Read more

## Infinite Cyclic Group is Isomorphic to Z [With Generators]

An infinite cyclic group is isomorphic to the additive group Z where Z is the set of integers. It has only two generators 1 and -1. In this article, we will study infinite cyclic groups along with its generators. A cyclic group is a group where every element is expressed as an integral power of … Read more

## Prove that Order of Element Divides Order of Group

The order of an element divides the order of a finite group. If G is a finite group and a ∈ G, then o(a) divides |G|. Let us now recall the definition of the order of an element in a group. Let a ∈ G. The order of a in G is the smallest positive integer n … Read more

## Subgroups of Cyclic Groups

A cyclic group G is a group where there exists a ∈ G such that every element of G can expressed as an integral powers of a, that is, if x ∈ G then x=an for some integer n. In this article, we will study subgroups of a cyclic group. Subgroups of Finite Cyclic Groups … Read more

## Every Subgroup of a Cyclic Group is Cyclic: Proof

Subgroup of any group always exists. If the group is cyclic, then its subgroup is also cyclic. In this article, we prove that each subgroup of a cyclic group is also cyclic. Let us first understand what are subgroups and cyclic groups. What are groups and subgroups? A set G forms a group with respect … Read more

## 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