Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Which of the following is /are true for group...
Start Learning for Free
Which of the following is /are true for group G?
- a)If the order of G in 2, then G is commutative
- b)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.
- c)If G is commutative then a subgroup of G need not to be commutative
- d)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of G
Correct answer is option 'A,B,D'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
Which of the following is /are true for group G?a)If the order of G in...
A. True
Theorem : All groups with less than 6 elements are abelian
B. True
To prove that G is an abelian group, we need ab = ba for any elements a.b in G.
By the given relation, we have (ab)2 = a2b2.
The left hand side is (ab)2 = (ab) (ab).
and thus the relation becomes (ab)2 = (ab) = a2b2.
Equivalently. we can express it as (ab) (ab) = aabb.
Multiplying by a-1 on the left and b-1011 the right, we obtain a ~1 (abab)b ~ 1 = a ~1 (aabb)b ~1 .
Since a-1a = e. bb-1 e. where e is the identity element of G. we have ebae = eabe.
Since e is the identity element, it yields that ba = ab
and this implies that G is an abelian group.
C. False
Just use proof by contradiction.
Suppose H is not abelian and thus contains two non-commuting members x and y. Then xy ^ yx. But x and y are also in G. and thus G is not abelian.
Contradiction.
D. True
Whenever we have a condition g2 = e in a group, it's equivalent to g = g-1 (multiply both sides by g-1).
In this case, it applies to every element of the group, so we can add remove inverses from any expression freely.
So the proof is simply ab (ab)-1 = b-1a-1 = ba.
Theorem : All groups with less than 6 elements are abelian
B. True
To prove that G is an abelian group, we need ab = ba for any elements a.b in G.
By the given relation, we have (ab)2 = a2b2.
The left hand side is (ab)2 = (ab) (ab).
and thus the relation becomes (ab)2 = (ab) = a2b2.
Equivalently. we can express it as (ab) (ab) = aabb.
Multiplying by a-1 on the left and b-1011 the right, we obtain a ~1 (abab)b ~ 1 = a ~1 (aabb)b ~1 .
Since a-1a = e. bb-1 e. where e is the identity element of G. we have ebae = eabe.
Since e is the identity element, it yields that ba = ab
and this implies that G is an abelian group.
C. False
Just use proof by contradiction.
Suppose H is not abelian and thus contains two non-commuting members x and y. Then xy ^ yx. But x and y are also in G. and thus G is not abelian.
Contradiction.
D. True
Whenever we have a condition g2 = e in a group, it's equivalent to g = g-1 (multiply both sides by g-1).
In this case, it applies to every element of the group, so we can add remove inverses from any expression freely.
So the proof is simply ab (ab)-1 = b-1a-1 = ba.
Free Test
FREE
| Start Free Test |
Community Answer
Which of the following is /are true for group G?a)If the order of G in...
Understanding the Properties of Group G
To determine the truth of the statements regarding group G, let’s analyze each option in detail:
Option A: If the order of G is 2, then G is commutative
- A group of order 2 has only two elements: the identity element and one other element.
- The multiplication table can only form one possible structure, which is trivial and commutative.
- Thus, this statement is **true**.
Option B: If for all x, y ∈ G, (xy)² = x²y², then G is commutative
- This condition implies that every product of elements behaves like a commutative operation.
- Expanding (xy)² gives us xyxy, and setting it equal to x²y² means that xyxy = xxyy, leading to the conclusion that xy = yx.
- Therefore, this statement is also **true**.
Option C: If G is commutative, then a subgroup of G need not be commutative
- If G is a commutative (abelian) group, all its subgroups inherit this property.
- Thus, if G is commutative, every subgroup must also be commutative, making this statement **false**.
Option D: If for all x ∈ G, x² = 1, then G is commutative
- This condition means that every element is its own inverse, which leads to the conclusion that the group must behave commutatively.
- For any x, y ∈ G, we have: (xy)² = xyxy = 1, which confirms xy = yx.
- Hence, this statement is **true**.
Conclusion
The correct statements regarding group G are A, B, and D.
To determine the truth of the statements regarding group G, let’s analyze each option in detail:
Option A: If the order of G is 2, then G is commutative
- A group of order 2 has only two elements: the identity element and one other element.
- The multiplication table can only form one possible structure, which is trivial and commutative.
- Thus, this statement is **true**.
Option B: If for all x, y ∈ G, (xy)² = x²y², then G is commutative
- This condition implies that every product of elements behaves like a commutative operation.
- Expanding (xy)² gives us xyxy, and setting it equal to x²y² means that xyxy = xxyy, leading to the conclusion that xy = yx.
- Therefore, this statement is also **true**.
Option C: If G is commutative, then a subgroup of G need not be commutative
- If G is a commutative (abelian) group, all its subgroups inherit this property.
- Thus, if G is commutative, every subgroup must also be commutative, making this statement **false**.
Option D: If for all x ∈ G, x² = 1, then G is commutative
- This condition means that every element is its own inverse, which leads to the conclusion that the group must behave commutatively.
- For any x, y ∈ G, we have: (xy)² = xyxy = 1, which confirms xy = yx.
- Hence, this statement is **true**.
Conclusion
The correct statements regarding group G are A, B, and D.
Attention Computer Science Engineering (CSE) Students!
To make sure you are not studying endlessly, EduRev has designed Computer Science Engineering (CSE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Computer Science Engineering (CSE).
|
Explore Courses for Computer Science Engineering (CSE) exam
|
|
Similar Computer Science Engineering (CSE) Doubts
Top Courses for Computer Science Engineering (CSE)View all
Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer?
Question Description
Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer?.
Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer?.
Solutions for Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Computer Science Engineering (CSE).
Download more important topics, notes, lectures and mock test series for Computer Science Engineering (CSE) Exam by signing up for free.
Here you can find the meaning of Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer?, a detailed solution for Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? has been provided alongside types of Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Which of the following is /are true for group G?a)If the order of G in 2, then G is commutativeb)If for all x, y ∈ G , (xy)2 = x2 y2, then G in commutative.c)If G is commutative then a subgroup of G need not to be commutatived)If for all x ∈ G , x2 = 1 , then G in commutative, here 1 in identity element of GCorrect answer is option 'A,B,D'. Can you explain this answer? tests, examples and also practice Computer Science Engineering (CSE) tests.
|
|
Explore Courses for Computer Science Engineering (CSE) exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup