# Thread: Find all the distinct cyclic subgroup of A4

1. ## Find all the distinct cyclic subgroup of A4

Find all the distinct cyclic subgroup of A4

what is distinct cyclic subgroup? Does it mean A4 = {1,2,3,4} ?
to find all of the subgroup, should it be: (1,2,3,4) , (2,1,3,4) , ( 1,2,4,3) etc ... or it should be (1,2) , (1,3), (1,4) ???

2. Originally Posted by rainyice
Find all the distinct cyclic subgroup of A4

what is distinct cyclic subgroup? Does it mean A4 = {1,2,3,4} ?
to find all of the subgroup, should it be: (1,2,3,4) , (2,1,3,4) , ( 1,2,4,3) etc ... or it should be (1,2) , (1,3), (1,4) ???
I hope you know that $A_4=\eta^{-1}(1)$ where $\eta:S_4\to \{-1,1\}:\sigma\mapsto\text{sgn }\sigma$. And then a cyclic subgroup of $A_4$ would be a subgroup that is generated by one element. What's the problem?

3. ## Still confused

since you said "then a cyclic subgroup of would be a subgroup that is generated by one element." Would that mean the answer is simply {(1), (2), (3), (4)} there are 4 distinct cyclic subgroups?

4. Originally Posted by natmov85
since you said "then a cyclic subgroup of would be a subgroup that is generated by one element." Would that mean the answer is simply {(1), (2), (3), (4)} there are 4 distinct cyclic subgroups?
Whose to say those are all distinct? I mean you could cover all of the cyclic subgroups of $G$ in the class $\left\{\langle g\rangle:g\in G\right\}$ but you wouldn't know which are distinct.

5. Originally Posted by natmov85
since you said "then a cyclic subgroup of would be a subgroup that is generated by one element." Would that mean the answer is simply {(1), (2), (3), (4)} there are 4 distinct cyclic subgroups?

List the different order out, and then find the cyclic subgroup. If you see any of them have the same cyclic subgroup, then just take one of them.