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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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Most Upvoted Answer
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:
  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.
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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:
  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.
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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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