# Abstract Algebra : Groups

## Example Questions

### Example Question #1 : Abstract Algebra

Which of the following is an identity element of the binary operation ?

Explanation:

Defining the binary operation  will help in understanding the identity element. Say  is a set and the binary operator is defined as  for all given pairs in .

Then there exists an identity element  in  such that given,

### Example Question #1 : Introduction

Which of the following illustrates the inverse element?

Explanation:

For every element in a set, there exists another element that when they are multiplied together results in the identity element.

In mathematical terms this is stated as follows.

For every  such that  where  and  is an identity element.

### Example Question #1 : Abstract Algebra

identify the following definition.

Given  is a normal subgroup of , it is denoted that  when the group of left cosets of  in  is called __________.

Subgroup

Cosets

Simple Group

Normal Group

Factor Group

Factor Group

Explanation:

By definition of a factor group it is stated,

Given  is a normal subgroup of , it is denoted that  when the group of left cosets of  in  is called the factor group of  which is determined by .

### Example Question #4 : Abstract Algebra

Determine whether the statement is true of false:

True

False