Cyclic Groups

[liamhatesmath]

Hi I'm having some trouble with this problem;

Is the following a cyclic group with 2 as a generator element?

(2ℤ, +)

I thought that the answer was yes but in the mark scheme it said that it wasn't. Could anyone explain to me why 2 isn't the generator?

Thanks

 

[Dr.Peterson]

Please quote the entire problem as given to you, and the entire statement in the mark scheme about the answer.

 

[liamhatesmath]

This is what the entire question looked like:

Which of the following are cyclic groups with 2 as a generator element? Select all correct answers.

(ℤ 9,+mod9)

({1,2,4,5,7,8},×mod9)

({2,4,6},×mod7)

(2ℤ ,+)

The correct answers were listed as only the first and the second option.

 

[pka]

Is the following a cyclic group with 2 as a generator element?
(2ℤ, +)
I thought that the answer was yes but in the mark scheme it said that it wasn't. Could anyone explain to me why 2 isn't the generator?

It occurs to me that the trouble may be with notation. The notation \(2\mathbb{Z}\) is not in my several books.

 

[liamhatesmath]

It occurs to me that the trouble may be with notation. The notation \(2\mathbb{Z}\) is not in my several books.

I think the notation is fine; I have found several papers online that include it. I found a section on isomorphic groups that stated:

As a group, (2ℤ , +) behaves identically to (ℤ , +). All we did was relabel the nonzero elements: rename 1 → 2 , 2 → 4 , etc.

 

[Dr.Peterson]

I have nothing more to say. Do you have an instructor you can ask?