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Statement: All cyclic groups are abelian. Statement B: The order of cyclic group is same as the order of its generator.
- a)A and B are false.
- b)A is true, B is false.
- c)B is true, A is false.
- d)A and B are true.
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Statement: All cyclic groups are abelian. Statement B: The order of cy...
Explanation:
The statement "All cyclic groups are abelian" is true because a cyclic group is a group that can be generated by a single element, called the generator. In a cyclic group, every element can be expressed as a power of the generator.
Proof:
Let G be a cyclic group generated by the element g. Then, for any two elements a and b in G, we can write them as powers of g: a = g^m and b = g^n, where m and n are integers.
To show that G is abelian, we need to prove that ab = ba for all a and b in G.
Using the expressions for a and b, we have:
ab = (g^m)(g^n) = g^(m+n)
ba = (g^n)(g^m) = g^(n+m)
Since addition of integers is commutative, m + n = n + m, so g^(m+n) = g^(n+m). Therefore, ab = ba, and G is abelian.
Statement B:
The statement "The order of a cyclic group is the same as the order of its generator" is also true.
Proof:
The order of an element a in a group is defined as the smallest positive integer n such that a^n = e, where e is the identity element.
In a cyclic group generated by an element g, the order of g is the smallest positive integer n such that g^n = e. This means that the order of the cyclic group is also n.
To prove this, we can show that the powers of g repeat after n steps.
Let's consider the powers of g: g, g^2, g^3, ..., g^n.
If we continue multiplying g by itself, we will eventually reach g^n+1. However, since the order of g is n, we know that g^n = e, so g^n+1 = g^1 = g.
Therefore, the powers of g repeat after n steps, and the order of the cyclic group is n, which is the same as the order of its generator.
Therefore, statement A and statement B are both true, and the correct answer is option D.
The statement "All cyclic groups are abelian" is true because a cyclic group is a group that can be generated by a single element, called the generator. In a cyclic group, every element can be expressed as a power of the generator.
Proof:
Let G be a cyclic group generated by the element g. Then, for any two elements a and b in G, we can write them as powers of g: a = g^m and b = g^n, where m and n are integers.
To show that G is abelian, we need to prove that ab = ba for all a and b in G.
Using the expressions for a and b, we have:
ab = (g^m)(g^n) = g^(m+n)
ba = (g^n)(g^m) = g^(n+m)
Since addition of integers is commutative, m + n = n + m, so g^(m+n) = g^(n+m). Therefore, ab = ba, and G is abelian.
Statement B:
The statement "The order of a cyclic group is the same as the order of its generator" is also true.
Proof:
The order of an element a in a group is defined as the smallest positive integer n such that a^n = e, where e is the identity element.
In a cyclic group generated by an element g, the order of g is the smallest positive integer n such that g^n = e. This means that the order of the cyclic group is also n.
To prove this, we can show that the powers of g repeat after n steps.
Let's consider the powers of g: g, g^2, g^3, ..., g^n.
If we continue multiplying g by itself, we will eventually reach g^n+1. However, since the order of g is n, we know that g^n = e, so g^n+1 = g^1 = g.
Therefore, the powers of g repeat after n steps, and the order of the cyclic group is n, which is the same as the order of its generator.
Therefore, statement A and statement B are both true, and the correct answer is option D.
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Community Answer
Statement: All cyclic groups are abelian. Statement B: The order of cy...
We are analyzing two statements about cyclic groups:
Statement A: All cyclic groups are abelian.
- A cyclic group is generated by a single element g, and every element in the group can be written as a power of g, i.e., gn.
- In a cyclic group, the group operation involves powers of g, and the order of multiplication does not matter because ga ⋅ gb = ga+b = gb ⋅ ga
- Thus, all cyclic groups are commutative (abelian).
- Statement A is true.
Statement B: The order of a cyclic group is the same as the order of its generator.
- The order of a group is the number of elements in it.
- The order of the generator g is the smallest positive integer n such that gn = e (identity element).
- In a cyclic group, the order of the generator g is exactly equal to the order of the group because g generates all elements of the group.
- Statement B is true.
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Statement: All cyclic groups are abelian. Statement B: The order of cyclic group is same as the order of its generator.a)A and B are false.b)A is true, B is false.c)B is true, A is false.d)A and B are true.Correct answer is option 'D'. Can you explain this answer?
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