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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 'B,C'. Can you explain this answer?
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Most Upvoted Answer
Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from...
We need to show that the centralizer of y, denoted as C(y), is a proper subgroup of G.
Assume, for the sake of contradiction, that C(y) = G. This means that for any g in G, we have gy = yg. Since G is nonabelian, there exists at least one element x in G such that x*y != y*x.
Let's consider the element x*y. Since C(y) = G, we know that x*y must be in G and it must commute with y. This means that (x*y)*y = y*(x*y), which simplifies to x*(y*y) = (y*x)*y. Since y*y = y (since y is in the center of G), this further simplifies to x*y = y*x, which contradicts our choice of x. Hence, our assumption that C(y) = G must be false.
Therefore, we have shown that C(y) is a proper subgroup of G.
Assume, for the sake of contradiction, that C(y) = G. This means that for any g in G, we have gy = yg. Since G is nonabelian, there exists at least one element x in G such that x*y != y*x.
Let's consider the element x*y. Since C(y) = G, we know that x*y must be in G and it must commute with y. This means that (x*y)*y = y*(x*y), which simplifies to x*(y*y) = (y*x)*y. Since y*y = y (since y is in the center of G), this further simplifies to x*y = y*x, which contradicts our choice of x. Hence, our assumption that C(y) = G must be false.
Therefore, we have shown that C(y) is a proper subgroup of G.
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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 'B,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 'B,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 'B,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 'B,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 'B,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 'B,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 'B,C'. Can you explain this answer?.
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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 'B,C'. Can you explain this answer? tests, examples and also practice IIT JAM tests.
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