Cyclic and non proper subgroups

[Bellarosa]

1. What is/are the condition for a group with no proper subgroup to be cyclic?

Homework Equations

3. this is just a general qustion I am asking in oder to prove something?

 

[HallsofIvy]

Let a be non-identity member of G. If for some n, aⁿ= e, then {a, a², ..., a^n-1, aⁿ= e} is a subgroup of G.

Now if G has no proper subgroups what is the smallest such n for any a? What does that tell you about G?

 

[Bellarosa]

I don't quite understand, but I'm guessing the smallest such n would be 1... can you give me an example of a Group with non proper subgroups

 

[Bellarosa]

I think I figurd it out ...the smallest of such n would be distinct , the group itself would bconsist of elements of infinite order... I still need an example of a Cyclic group with non proper subgroups

 

[HallsofIvy]

Do you really understand what you are asking? Any group of prime order has no proper subgroup.

 

[Bellarosa]

Not quite...Ok this is my problem:
If G has noroper subgroups, prove that G is cyclic.

Proof:
If G has no proper subgroup then |G|= p. For any nonidentity element a belonging to G, <a> is a subgroup of order greater than 1. By Langrange's Theorem, since |a|divides |G| |a| = p therefore, <a> = G and G is a cyclic group of order p.

Does this proof make sense? In your first question I sain that the smallest scuh n is 1 hence the reason why I said that |a| must be greater than one...I'm notre if my proof is correct but does it make sense?