Question

Given:

Statement A: All cyclic groups are an abelian group.

Statement B: The order of the cyclic group is the same as the order of its generator.

1. A and B are false
2. A is true, B is false
3. B is true, A is false
4. A and B both are true

Answer 1

Correct Answer - Option 4 : A and B both are true

Concept:

Abelian Group: Let {G=e, a, b} where e is identity. The operation 'o' is defined by the following composition table. Then(G, o) is called Abelian if it follows the following property-

1. Closure Property
2. Associativity
3. Existence of Identity
4. Existence of Inverse
5. Commutativity

Cyclic Group- A group a is said to be cyclic if it contains an element 'a' such that every element of G can be represented as some integral power of 'a'. The element 'a' is then called a generator of G, and G is denoted by <a> (or [a]). 

Theorem: 

(i) All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. 

(ii) The order of a cyclic group is the same as the order of its generator. 

Thus it is clear that A and B both are true.