Consider the given statements:Statement A: All cyclic groups are abelian groups.Statement B: The order of the cyclic group is the same as the order of its generator.Which of these are true/false?a)A and B are falseb)A is true, B is falsec)B is true, A is falsed)A and B both are trueCorrect answer is option 'D'. Can you explain this answer? - EduRev Computer Science Engineering (CSE) Question

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Consider the given statements:

Statement A: All cyclic groups are abelian groups.

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

Which of these are true/false?

a)
A and B are false
b)
A is true, B is false
c)
B is true, A is false
d)
A and B both are true

Correct answer is option 'D'. Can you explain this answer?

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Consider the given statements:Statement A: All cyclic groups are abeli...

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:

Closure Property
Associativity
Existence of Identity
Existence of Inverse
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.

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Consider the given statements:Statement A: All cyclic groups are abeli...

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:

Closure Property
Associativity
Existence of Identity
Existence of Inverse
Commutativity

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.

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Consider the given statements:Statement A: All cyclic groups are abelian groups.Statement B: The order of the cyclic group is the same as the order of its generator.Which of these are true/false?a)A and B are falseb)A is true, B is falsec)B is true, A is falsed)A and B both are trueCorrect answer is option 'D'. Can you explain this answer?

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