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Let G be an infinite cyclic group and H is its subgroup. Which one of the following is correct ?
- a)H is not necessarily cyclic.
- b)H is finite.
- c)H is infinite.
- d)H is not necessarily abelian.
Correct answer is option 'C'. Can you explain this answer?
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Let G be an infinite cyclic group and H is its subgroup. Which one of ...
Cyclic group - A group G is said to be cyclic, if, for some a ∈ G, every element x ∈ G is of the form an, where n is some integers. The element a is called a generator of G.
There may be more than one generators of a cyclic group. If G is a cyclic group generated by a, then we shall write G = {a} or G = (a). The elements of G will be of the form
..., a-3, a-2, a-1, a° = e, a, a2, a3, ...
Some properties
(i) Every cyclic group is an abelian group.
(ii) Every subgroup of a cyclic group is cyclic.
(iii) Every proper subgroup of an infinite cyclic group is infinite.
Hence by property III, if H is a subgroup of an infinite cyclic group, then H is a also infinite.
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Let G be an infinite cyclic group and H is its subgroup. Which one of ...
Cyclic group - A group G is said to be cyclic, if, for some a ∈ G, every element x ∈ G is of the form an, where n is some integers. The element a is called a generator of G.
There may be more than one generators of a cyclic group. If G is a cyclic group generated by a, then we shall write G = {a} or G = (a). The elements of G will be of the form
..., a-3, a-2, a-1, a° = e, a, a2, a3, ...
Some properties
(i) Every cyclic group is an abelian group.
(ii) Every subgroup of a cyclic group is cyclic.
(iii) Every proper subgroup of an infinite cyclic group is infinite.
Hence by property III, if H is a subgroup of an infinite cyclic group, then H is a also infinite.
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Let G be an infinite cyclic group and H is its subgroup. Which one of ...
Cyclic group - A group G is said to be cyclic, if, for some a ∈ G, every element x ∈ G is of the form an, where n is some integers. The element a is called a generator of G.
There may be more than one generators of a cyclic group. If G is a cyclic group generated by a, then we shall write G = {a} or G = (a). The elements of G will be of the form
..., a-3, a-2, a-1, a° = e, a, a2, a3, ...
Some properties
(i) Every cyclic group is an abelian group.
(ii) Every subgroup of a cyclic group is cyclic.
(iii) Every proper subgroup of an infinite cyclic group is infinite.
Hence by property III, if H is a subgroup of an infinite cyclic group, then H is a also infinite.
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Let G be an infinite cyclic group and H is its subgroup. Which one of the following is correct ?a)H is not necessarily cyclic.b)H is finite.c)H is infinite.d)H is not necessarily abelian.Correct answer is option 'C'. Can you explain this answer?
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