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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:

e*b=eb

b*b=e

e*b=b

e*b=e^2

e*b=b^2

Correct answer:

e*b=b

Explanation:

Defining the binary operation (*) will help in understanding the identity element. Say is a set and the binary operator is defined as $A\times A\rightarrow A$ for all given pairs in .

Then there exists an identity element in such that given,

$\\a,b,c \ \epsilon\ A$

$\\a*e=a, e*a=a \\b*e=b, e*b=b \\c*e=c,e*c=c$

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

e*b=b

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Example Question #2 : Abstract Algebra

Which of the following illustrates the inverse element?

Possible Answers:

b*e=b

a*b=b

a*e=a

a*b=a

a*b=e

Correct answer:

a*b=e

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 $a\ \epsilon\ S\ \exists\ b\ \epsilon\ S$ such that a*b=e where $e\ \epsilon\ S$ and is an identity element.

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Example Question #3 : Abstract Algebra

identify the following definition.

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

Possible Answers:

Factor Group

Cosets

Subgroup

Simple Group

Normal 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 A/B when the group of left cosets of in is called the factor group of which is determined by .

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Example Question #3 : Abstract Algebra

Determine whether the statement is true of false:

(aA)(bA)=abA

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,

(aA)(bA)=abA

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Abstract Algebra : Groups

Study concepts, example questions & explanations for Abstract Algebra

All Abstract Algebra Resources

Example Questions

Example Question #1 : Abstract Algebra

Example Question #2 : Abstract Algebra

Example Question #3 : Abstract Algebra

Example Question #3 : Abstract Algebra

All Abstract Algebra Resources

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