# Cyclic and non proper subgroups

1. Nov 12, 2008

### Bellarosa

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

2. Relevant equations

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

2. Nov 13, 2008

### HallsofIvy

Staff Emeritus
Let a be non-identity member of G. If for some n, an= e, then {a, a2, ..., an-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?

3. Nov 14, 2008

### 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

4. Nov 14, 2008

### 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

5. Nov 14, 2008

### HallsofIvy

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

6. Nov 15, 2008

### 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?