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Check the correct statement.
- a)Every subgroup of a cyclic group is cyclic.
- b)If G is an infinite cyclic group, then G has exactly two generators and G is isomorphic to the additive group of integers.
- c)Every finite group of composite order possesses proper subgroups.
- d)All of the above.
Correct answer is option 'D'. Can you explain this answer?
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Check the correct statement.a)Every subgroup of a cyclic group is cycl...
Explanation:
The correct statement is option 'D': All of the above. Let's break down each statement and explain why it is correct.
a) Every subgroup of a cyclic group is cyclic:
This statement is true. A cyclic group is generated by a single element, and therefore, every subgroup of a cyclic group is also generated by a single element. Hence, every subgroup of a cyclic group is cyclic.
b) If G is an infinite cyclic group, then G has exactly two generators and G is isomorphic to the additive group of integers:
This statement is also true. An infinite cyclic group is generated by a single element, say 'a'. Any other generator of the group can be written as 'a^n' for some integer n. Therefore, there are exactly two generators: 'a' and 'a^(-1)' (the inverse of 'a'). Moreover, the additive group of integers under addition is isomorphic to an infinite cyclic group.
c) Every finite group of composite order possesses proper subgroups:
This statement is true as well. A composite number is a positive integer greater than 1 that is not a prime number. If a finite group has composite order, then it can be written as the product of two or more prime numbers. According to the Fundamental Theorem of Finite Abelian Groups, a group of composite order can be decomposed into a direct product of cyclic groups of prime power order. Each of these cyclic groups is a proper subgroup of the original group. Hence, every finite group of composite order possesses proper subgroups.
Therefore, all the given statements are correct, and the correct answer is option 'D': All of the above.
The correct statement is option 'D': All of the above. Let's break down each statement and explain why it is correct.
a) Every subgroup of a cyclic group is cyclic:
This statement is true. A cyclic group is generated by a single element, and therefore, every subgroup of a cyclic group is also generated by a single element. Hence, every subgroup of a cyclic group is cyclic.
b) If G is an infinite cyclic group, then G has exactly two generators and G is isomorphic to the additive group of integers:
This statement is also true. An infinite cyclic group is generated by a single element, say 'a'. Any other generator of the group can be written as 'a^n' for some integer n. Therefore, there are exactly two generators: 'a' and 'a^(-1)' (the inverse of 'a'). Moreover, the additive group of integers under addition is isomorphic to an infinite cyclic group.
c) Every finite group of composite order possesses proper subgroups:
This statement is true as well. A composite number is a positive integer greater than 1 that is not a prime number. If a finite group has composite order, then it can be written as the product of two or more prime numbers. According to the Fundamental Theorem of Finite Abelian Groups, a group of composite order can be decomposed into a direct product of cyclic groups of prime power order. Each of these cyclic groups is a proper subgroup of the original group. Hence, every finite group of composite order possesses proper subgroups.
Therefore, all the given statements are correct, and the correct answer is option 'D': All of the above.
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Check the correct statement.a)Every subgroup of a cyclic group is cyclic.b)If G is an infinite cyclic group, then G has exactly two generators and G is isomorphic to the additive group of integers.c)Every finite group of composite order possesses proper subgroups.d)All of the above.Correct answer is option 'D'. Can you explain this answer?
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Solutions for Check the correct statement.a)Every subgroup of a cyclic group is cyclic.b)If G is an infinite cyclic group, then G has exactly two generators and G is isomorphic to the additive group of integers.c)Every finite group of composite order possesses proper subgroups.d)All of the above.Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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