Abstract Algebra : Groups

Study concepts, example questions & explanations for Abstract Algebra

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Example Questions

Example Question #1 : Abstract Algebra

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

Possible Answers:

Correct answer:

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,

Therefore, looking at the possible answer selections the correct answer is,

Example Question #1 : Introduction

Which of the following illustrates the inverse element?

Possible Answers:

Correct answer:

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 __________.

Possible Answers:

Subgroup

Cosets

Simple Group

Normal Group

Factor Group

Correct answer:

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:

Possible Answers:

True

False

Correct answer:

True

Explanation:

This statement is true based on the following theorem.

For all  in .

If  is a normal subgroup of  then the cosets of  forms a group under the multiplication given by,

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