Testbook set theory question for abelian group and generators

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

Comments

I didn’t get the second option

Answer 1

Statement 1 is correct  as It is a standard result and proof is quite simple .

Proof:- https://math.stackexchange.com/questions/44995/are-cyclic-groups-always-abelian

Statement 2:- It is false . Why ?

The correct statement would be : –  ‘’ The order of the finite cyclic group is same as the order of its generator . ‘’ The word “Finite ” is not mentioned .

why in case of infinite group this problem would not satisfied ?

=> Let take the example for this group $G=$ $\left ( \mathbb{Z},+ \right )$ .

This is a infinite cyclic group which has 2 generators $1$ and $-1$ . 

So order of $G$ is Infinite .

Now order of generator is also infinite as there does not exist any n such that  $a^{n}=e$ where $a$ is a generator and $e$ is identity element and $n\in \mathbb{Z}$ for group G.

So order of G is Infinite and order of its generator also infinite . So now can we compare two infinite to check if they equal not ? 

=> We can’t .

So for infinite group the result does not hold .

So correct option is option (2) .

proof :- https://math.stackexchange.com/questions/595644/in-a-finite-cyclic-group-the-generator-has-the-order-of-the-group

 

Comments

Bro answer given if option 4 .

Statement second is given True.

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It should be true finite group .I believe answer given is wrong .

@Deepak Poonia  sir pleasee confirm this.

Answer 2

I am adding what explanation is given :

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.

Comments

According to them :

The Order of a cyclic group is same as order of its generator .

I didn’t get any proof there though.