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Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.
Then
Then
- a)g and h are homomorphisms and f is not a homomorphism
- b)h is a homomorphism and g is not a homomorphism
- c)f is a homomorphism and g is not a homomorphism
- d)f, g and h are homomorphisms
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from...
Let G be a nonabelian group, and let y be an element of G.
Since G is nonabelian, there exist elements a and b in G such that ab is not equal to ba.
Consider the element x = ayba^(-1).
We want to show that x is not equal to y.
Assume for contradiction that x = y.
Then, we have ayba^(-1) = y.
Multiplying both sides on the right by a, we get ayba^(-1)a = ya.
Simplifying, we have ayb = ya.
Multiplying both sides on the right by b^(-1), we get aybb^(-1) = yab^(-1).
Simplifying, we have ay = yab^(-1).
Multiplying both sides on the left by a^(-1), we get a^(-1)ay = a^(-1)yab^(-1).
Simplifying, we have y = a^(-1)yab^(-1).
But this implies that ab = ba, which contradicts the assumption that G is nonabelian.
Therefore, our assumption that x = y is false, and we conclude that x is not equal to y.
Since G is nonabelian, there exist elements a and b in G such that ab is not equal to ba.
Consider the element x = ayba^(-1).
We want to show that x is not equal to y.
Assume for contradiction that x = y.
Then, we have ayba^(-1) = y.
Multiplying both sides on the right by a, we get ayba^(-1)a = ya.
Simplifying, we have ayb = ya.
Multiplying both sides on the right by b^(-1), we get aybb^(-1) = yab^(-1).
Simplifying, we have ay = yab^(-1).
Multiplying both sides on the left by a^(-1), we get a^(-1)ay = a^(-1)yab^(-1).
Simplifying, we have y = a^(-1)yab^(-1).
But this implies that ab = ba, which contradicts the assumption that G is nonabelian.
Therefore, our assumption that x = y is false, and we conclude that x is not equal to y.
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Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer?
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Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer?.
Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer?.
Solutions for Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
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Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.Thena)g and h are homomorphisms and f is not a homomorphismb)h is a homomorphism and g is not a homomorphismc)f is a homomorphism and g is not a homomorphismd)f, g and h are homomorphismsCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice IIT JAM tests.
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