Contents

- 1Group
- 1.1Closure
- 1.2Associativity
- 1.3Identity
- 1.4Inverse
- 1.5Some Examples of Group
- 1.6Example 1
- 1.7Example 2
- 1.8Subgroup
- 1.9Group Table
- 2Homomorphism of a Group
- 2.1Examples of Group Homomorphism
- 2.1.1Example 1
- 2.2Endomorphism
- 2.2.1Examples of Endomorphism
- 2.2.1.1Example 1
- 2.3Isomorphism
- 2.3.1Examples of Isomorphism
- 2.3.1.1Example 1
- 2.4Automorphism
- 2.4.1Examples of Automorphism
- 2.4.1.1Example 1
- 3Kernel of Homomorphism
- 3.1Examples of Kernel of homomorphism
- 3.1.1Example 1
- 3.1.2Example 2
- 3.1.3Example 3
- 3.1.4Example 4
- 4Some simplified theorems derived from homomorphism
- 4.1Theorem 1
- 4.1.1Example for Theorem 1
- 4.2Theorem 2
- 4.3Theorem 3
- 4.4Theorem 4
- 4.5Theorem 5
- 4.6Theorem 6

# Group

A group is any set G with a defined binary operation (called the **group law** of ), written as 2 tuple (examples: ), satisfying 4 basic rules

## Closure

The important point to be understood about a binary operation on is that is closed with respect to in the sense that if then

( can be read as "**a,b element of C**" or "**a,b in C**")

## Associativity

( can be read as "**for all a,b in C**" or "**for all a,b being element of C**" or "**for each a,b in C**" or "**for every a,b in C**", ... etc.)

## Identity

An element (called **identity** of the Group ) that satisfies the condition

contains at most one identity element

## Inverse

there exists an element such that

*Groups can be both finite and infinite.*

## Some Examples of Group

## Example 1

The set (set of all real numbers excluding ) with the binary operation of multiplication forms a group.

**closure criteria**

For instance,

5*6 =30 is an element of

**associativity criteria**

For instance,

2*(3*9)=(2*3)*9 = 2*3*9=54

**identity Criteria**

number is identity for in

For instance,

1*5=5*1=5

**inverse criteria**

For instance,

## Example 2

The set with the binary operation of addition forms another group

**closure criteria**

For instance,

5+6 =11 is an element of

**associativity criteria**

For instance,

2+(3+9)=(2+3)+9 = 2+3+9=14

**identity Criteria**

number is identity for in

For instance,

0+5=5+0=5

**inverse criteria**

For instance,

## Subgroup

Given a group under a binary operation , a subset of is called a subgroup of if also forms a group under the operation

Both Group() and Subgroup() share the same identity .

## Group Table

Group table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table (reminiscent of an addition or multiplication table). Many properties of a group - such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group - can be discovered from group table

For group the multiplication table looks like:

For group , the multiplication table looks like:

# Homomorphism of a Group

if and are two groups with binary operations and , respectively, a function is a homomorphism if

,

Simply put, group homomorphism is a transformation of one Group into another that preserves (invariant) in the second Group the relations between elements of the first.

## Examples of Group Homomorphism

### Example 1

Let be the group of all **nonsingular**, real, matrices with the binary operation of matrix multiplication. Let be the group with the binary operation of scalar multiplication. The function that is the determinant of a matrix is then a homomorphism from to

To put this in symbolic context:

Let and let

Then,

## Endomorphism

A homomorphism is called an endomorphism

### Examples of Endomorphism

#### Example 1

Let be the group with the binary operation of multiplication. The function that takes the absolute value of a number is then an endomorphism of into

To put this in symbolic context:

Let , Then,

,

For instance,

|-3*4|=|-3|*|4|=3*4=12

And

|-2| = |2|=2

## Isomorphism

A homomorphism is an isomorphism if is both **one-to-one** and **onto** (**bijective**).

### Examples of Isomorphism

#### Example 1

Let be the group of positive real numbers with the binary operation of multiplication and let be the group of real numbers with the binary operation of addition. The function is an isomorphism between and

To put this in symbolic context:

Let , and Let , ,

Then,

,

( denotes set of all positive real numbers)

## Automorphism

An isomorphism is called an automorphism.

### Examples of Automorphism

#### Example 1

Let be the group with the binary operation of multiplication. The function that takes the absolute value of a number is then an automorphism of into

To put this in symbolic context:

Let , , Then, ,

For instance,

[3*4|=|3|*|4|=3*4=12

In this case both sides can use only positive real numbers. Note that this contrasts with an earlier example of Endomorphism where the Group

# Kernel of Homomorphism

The kernel of a homomorphism is the subgroup of .

In other words, the kernel of is the set of elements of that are mapped by to the identity element of

The notation can be used to denote the kernel of

## Examples of Kernel of homomorphism

### Example 1

Let be the group of all nonsingular, real, matrices with the binary operation of matrix multiplication. Let be the group with the binary operation of scalar multiplication. The function that is the determinant of a matrix is then a homomorphism from to

Let and let

Then,

In this case the kernel of consists of set of all matrices with determinant equal to the real number

### Example 2

Let , , Then,

,

The kernel of consists of set

### Example 3

Let , , Then,

,

The kernel of consists of set

### Example 4

Let , and Let ,

Then,

,

The kernel of is the set because identity of is mapped to set of all numbers whose produces

# Some simplified theorems derived from homomorphism

(Note: I will not provide any proofs here, because the theorems are quiet simple and proofs can be worked out easily)

## Theorem 1

If is a homomorphism, then coincides with the identity element of and

### Example for Theorem 1

Let be the group of all nonsingular, real, matrices with the binary operation of matrix multiplication. Let be the group with the binary operation of scalar multiplication. The function that is the determinant of a matrix is then a homomorphism from to

To put this in symbolic context:

Let and let

Then,

In this case the identity of Matrix is the **Identity matrix** (denoted by ).

Therefore, , where: and is identity of

Now For , Let

Then,

&

We can verify that,

## Theorem 2

If is a homomorphism and if is a subgroup of , then is a subgroup of H

## Theorem 3

If is a homomorphism and if is a subgroup of , then the preimage is a subgroup of

## Theorem 4

A homomorphism is one-to-one if and only if .

## Theorem 5

If is an isomorphism, then is an isomorphism

## Theorem 6

A homomorphism is an isomorphism if it is onto and if its kernel contains only the identity element of G.