
Exploring Group Theory
Quiz
•
Mathematics
•
University
•
Practice Problem
•
Easy
Jili Basumatary
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20 questions
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1.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
What are the four properties that define a group?
Closure, Associativity, Identity Element, Invertibility
Distributivity
Commutativity
Associative Property
2.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
Given the set of integers under addition, is it a group? Why or why not?
No, the set of integers under addition is not a group because it lacks an identity element.
No, because the set of integers does not include inverses for all elements.
Yes, the set of integers under addition is a group.
Yes, but only for positive integers under addition.
3.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
If G is a group and a, b are elements of G, what is the result of the operation a * b?
a * b is an element of G.
a * b is equal to b * a.
a * b is an element of G but not unique.
a * b is not defined in G.
4.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
Explain the concept of the identity element in a group.
The identity element is the largest element in the group.
The identity element is the element that can be combined with itself to produce any other element.
The identity element is the element that has no effect when combined with itself.
The identity element in a group is an element that, when combined with any element of the group, does not change that element.
5.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
What is the inverse of an element in a group? Provide an example.
In the group of natural numbers under addition, the inverse of 3 is 0.
In the group of real numbers under addition, the inverse of 3 is 3.
In the group of integers under multiplication, the inverse of 3 is 1/3.
For example, in the group of integers under addition, the inverse of 3 is -3, since 3 + (-3) = 0, where 0 is the identity element.
6.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
How do you determine if a subset of a group is also a group?
A subset H of a group G is a group if it has at least one element.
A subset H of a group G is a group if it is finite in size.
A subset H of a group G is a group if it satisfies closure, contains the identity element, and includes inverses for all its elements.
A subset H of a group G is a group if it contains all elements of G.
7.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
What is a cyclic group? Give an example.
An example of a cyclic group is the group of integers under addition, denoted as (Z, +), where any integer can be generated by adding 1 (or -1) repeatedly.
The group of rational numbers under multiplication, denoted as (Q, *)
The group of non-zero real numbers under addition, denoted as (R*, +)
The set of all even integers under addition, denoted as (2Z, +)
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