Mathematics Exam > Mathematics Questions > Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is i...
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Let H = Z2 x Z6 and K = Z2 x Z4, then
- a)H is isomorphic to K, since both are cyclic.
- b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1
- c)H is not isomorphic to K, since there is no homomorphism from H to K.
- d)H is not isomorphic to K, since K is cyclic where as H is not.
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since bot...
Explanation:
Isomorphism between groups:
An isomorphism between two groups is a bijective homomorphism. It means that two groups are isomorphic if there exists a function between them that preserves the group structure, and is both one-to-one and onto.
Types of groups:
In this question, we have two groups:
- Group H = Z2 x Z6: This group is the direct product of Z2 and Z6. It consists of all ordered pairs (a, b), where a is an element of Z2 and b is an element of Z6.
- Group K = Z2 x Z4: This group is the direct product of Z2 and Z4. It consists of all ordered pairs (c, d), where c is an element of Z2 and d is an element of Z4.
Isomorphism between H and K:
To determine if H and K are isomorphic, we need to find a bijective homomorphism between them.
Contradiction in Option A:
Option A states that H is isomorphic to K because both groups are cyclic. However, this statement is incorrect because not all cyclic groups are isomorphic. In this case, H is not cyclic as it is the direct product of two different cyclic groups.
Contradiction in Option B:
Option B states that H is isomorphic to K because 2 divides 6 and gcd(3, 4) = 1. However, this statement is also incorrect. While it is true that 2 divides 6 and gcd(3, 4) = 1, these conditions alone do not guarantee isomorphism between the groups.
Correct explanation in Option C:
Option C states that H is not isomorphic to K because K is cyclic whereas H is not. This statement is correct. The group K = Z2 x Z4 is cyclic because it can be generated by a single element. However, the group H = Z2 x Z6 is not cyclic as it cannot be generated by a single element. Therefore, H and K cannot be isomorphic.
No homomorphism in Option D:
Option D states that H is not isomorphic to K because there is no homomorphism from H to K. This statement is also correct. Since H and K have different structures and properties, there cannot exist a homomorphism that preserves the group structure between them.
Conclusion:
In conclusion, option C is the correct answer. H is not isomorphic to K because K is cyclic whereas H is not. Additionally, there is no homomorphism from H to K.
Isomorphism between groups:
An isomorphism between two groups is a bijective homomorphism. It means that two groups are isomorphic if there exists a function between them that preserves the group structure, and is both one-to-one and onto.
Types of groups:
In this question, we have two groups:
- Group H = Z2 x Z6: This group is the direct product of Z2 and Z6. It consists of all ordered pairs (a, b), where a is an element of Z2 and b is an element of Z6.
- Group K = Z2 x Z4: This group is the direct product of Z2 and Z4. It consists of all ordered pairs (c, d), where c is an element of Z2 and d is an element of Z4.
Isomorphism between H and K:
To determine if H and K are isomorphic, we need to find a bijective homomorphism between them.
Contradiction in Option A:
Option A states that H is isomorphic to K because both groups are cyclic. However, this statement is incorrect because not all cyclic groups are isomorphic. In this case, H is not cyclic as it is the direct product of two different cyclic groups.
Contradiction in Option B:
Option B states that H is isomorphic to K because 2 divides 6 and gcd(3, 4) = 1. However, this statement is also incorrect. While it is true that 2 divides 6 and gcd(3, 4) = 1, these conditions alone do not guarantee isomorphism between the groups.
Correct explanation in Option C:
Option C states that H is not isomorphic to K because K is cyclic whereas H is not. This statement is correct. The group K = Z2 x Z4 is cyclic because it can be generated by a single element. However, the group H = Z2 x Z6 is not cyclic as it cannot be generated by a single element. Therefore, H and K cannot be isomorphic.
No homomorphism in Option D:
Option D states that H is not isomorphic to K because there is no homomorphism from H to K. This statement is also correct. Since H and K have different structures and properties, there cannot exist a homomorphism that preserves the group structure between them.
Conclusion:
In conclusion, option C is the correct answer. H is not isomorphic to K because K is cyclic whereas H is not. Additionally, there is no homomorphism from H to K.
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Community Answer
Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since bot...
For cyclic in external products group a condition must satisfy that is G=Zn×Zm is cyclic iff GCD(n, m)=1 so option here H is abelian but not cyclic where as K is cyclic and we have a property where it states every cyclic group is abelian but converse is not true.... example K4 is abelian but not cyclic.. so we can conclude that H and K are not isomorphic
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Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since both are cyclic.b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1c)H is not isomorphic to K, since there is no homomorphism from H to K.d)H is not isomorphic to K, since K is cyclic where as H is not.Correct answer is option 'C'. Can you explain this answer?
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Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since both are cyclic.b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1c)H is not isomorphic to K, since there is no homomorphism from H to K.d)H is not isomorphic to K, since K is cyclic where as H is not.Correct answer is option 'C'. 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 Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since both are cyclic.b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1c)H is not isomorphic to K, since there is no homomorphism from H to K.d)H is not isomorphic to K, since K is cyclic where as H is not.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since both are cyclic.b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1c)H is not isomorphic to K, since there is no homomorphism from H to K.d)H is not isomorphic to K, since K is cyclic where as H is not.Correct answer is option 'C'. Can you explain this answer?.
Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since both are cyclic.b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1c)H is not isomorphic to K, since there is no homomorphism from H to K.d)H is not isomorphic to K, since K is cyclic where as H is not.Correct answer is option 'C'. 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 Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since both are cyclic.b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1c)H is not isomorphic to K, since there is no homomorphism from H to K.d)H is not isomorphic to K, since K is cyclic where as H is not.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since both are cyclic.b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1c)H is not isomorphic to K, since there is no homomorphism from H to K.d)H is not isomorphic to K, since K is cyclic where as H is not.Correct answer is option 'C'. Can you explain this answer?.
Solutions for Let H = Z2 x Z6 and K = Z2 x Z4, thena)H is isomorphic to K, since both are cyclic.b)H is isomorphic to K since 2 divides 6 and gcd(3,4) = 1c)H is not isomorphic to K, since there is no homomorphism from H to K.d)H is not isomorphic to K, since K is cyclic where as H is not.Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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