Abstract Algebra : Structure Theory

Study concepts, example questions & explanations for Abstract Algebra

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Example Questions

Example Question #1 : Groups

identify the following definition. 

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

Possible Answers:

Subgroup

Normal Group

Factor Group

Simple Group

Cosets

Correct answer:

Factor Group

Explanation:

By definition of a factor group it is stated,

Given \(\displaystyle A\) is a normal subgroup of \(\displaystyle B\), it is denoted that \(\displaystyle A/B\) when the group of left cosets of \(\displaystyle A\) in \(\displaystyle B\) is called the factor group of \(\displaystyle B\) which is determined by \(\displaystyle A\).

Example Question #1 : Abstract Algebra

Determine whether the statement is true of false:

\(\displaystyle (aA)(bA)=abA\)

Possible Answers:

True

False

Correct answer:

True

Explanation:

This statement is true based on the following theorem.

For all \(\displaystyle a\)\(\displaystyle b\) in \(\displaystyle G\).

If \(\displaystyle A\) is a normal subgroup of \(\displaystyle G\) then the cosets of \(\displaystyle A\) forms a group under the multiplication given by,

\(\displaystyle (aA)(bA)=abA\)

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