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?

Question

Answer

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.

Similar Mathematics Doubts

Which one of the following is false

Consider the statements:S1: Let G be an abelian group of order n if for every divisior m of n there exist a subgroup of G of order m, thenG is cyclic.S2,: Let G be a group. If every proper sub group of G is cyclic then G is abelian.Which of the following is true.

Choose the correct option

Which statements is/are true ?

Choose the correct statements;

Top Courses for Mathematics

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams