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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?
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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.
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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.
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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?
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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?.
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