Mathematics Exam > Mathematics Questions > Which one of the following conditions on a gr...
Start Learning for Free
Which one of the following conditions on a group G implies that G is abelian?
- a)The order of G is p3 for some prime p
- b)Every proper subgroup of G is cyclic
- c)Every subgroup of G is normal in G
- d)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphism
Correct answer is option 'D'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
Which one of the following conditions on a group G implies that G is a...
Explanation:
To prove that option 'D' implies that G is abelian, we need to show that if the function f : G -> G defined by f(x) = x^1 for all x in G is a homomorphism, then G is abelian.
1. The function f is a homomorphism:
To prove that f is a homomorphism, we need to show that f(xy) = f(x)f(y) for all x, y in G.
Let x, y be any elements in G. Then,
f(xy) = (xy)^1 = xy
f(x)f(y) = x^1y^1 = xy
Since f(xy) = f(x)f(y) for all x, y in G, we can conclude that f is a homomorphism.
2. Every homomorphism from G to G is given by conjugation:
It can be proven that every homomorphism from G to G is given by conjugation, i.e., for any homomorphism h : G -> G, there exists an element a in G such that h(x) = axa^(-1) for all x in G.
3. Every element in G commutes with every other element:
From the given information, we know that the function f : G -> G defined by f(x) = x^1 for all x in G is a homomorphism. From point 2, we can conclude that f(x) = axa^(-1) for all x in G, where a is some element in G.
Since f(x) = x^1 = x, we have x = axa^(-1) for all x in G.
This implies that every element in G commutes with every other element, i.e., G is abelian.
Therefore, option 'D' implies that G is abelian.
Conclusion:
The condition that the function f : G -> G defined by f(x) = x^1 for all x in G is a homomorphism implies that G is abelian.
To prove that option 'D' implies that G is abelian, we need to show that if the function f : G -> G defined by f(x) = x^1 for all x in G is a homomorphism, then G is abelian.
1. The function f is a homomorphism:
To prove that f is a homomorphism, we need to show that f(xy) = f(x)f(y) for all x, y in G.
Let x, y be any elements in G. Then,
f(xy) = (xy)^1 = xy
f(x)f(y) = x^1y^1 = xy
Since f(xy) = f(x)f(y) for all x, y in G, we can conclude that f is a homomorphism.
2. Every homomorphism from G to G is given by conjugation:
It can be proven that every homomorphism from G to G is given by conjugation, i.e., for any homomorphism h : G -> G, there exists an element a in G such that h(x) = axa^(-1) for all x in G.
3. Every element in G commutes with every other element:
From the given information, we know that the function f : G -> G defined by f(x) = x^1 for all x in G is a homomorphism. From point 2, we can conclude that f(x) = axa^(-1) for all x in G, where a is some element in G.
Since f(x) = x^1 = x, we have x = axa^(-1) for all x in G.
This implies that every element in G commutes with every other element, i.e., G is abelian.
Therefore, option 'D' implies that G is abelian.
Conclusion:
The condition that the function f : G -> G defined by f(x) = x^1 for all x in G is a homomorphism implies that G is abelian.
Free Test
FREE
| Start Free Test |
Community Answer
Which one of the following conditions on a group G implies that G is a...
Given that function mapping is homomorphism so it is one - one so G is abelian.
|
Explore Courses for Mathematics exam
|
|
Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer?
Question Description
Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer?.
Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer?.
Solutions for Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Which one of the following conditions on a group G implies that G is abelian?a)The order of G is p3 for some prime pb)Every proper subgroup of G is cyclicc)Every subgroup of G is normal in Gd)The function f : G → G, defined by f(x) = x–1 for all x ∈ G, is a homomorphismCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics 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
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup