Isomorphism in Groups with No Proper Subgroups and Absolute Value Greater than 1

[kathrynag]

Homework Statement

Let G be any group with no proper , nontrivial subgroups and assume abs value(G)>1. Prove that G must be isomorphic to Z_p for some prime p.

Homework Equations

The Attempt at a Solution

I know we have an isomorphism if a group is 1-1, onto, and the homomorphism property holds.

 

[Tedjn]

What happens if G is cyclic? In this case, if G does not contain p elements, can you find a nontrivial subgroup? What happens if G is not cyclic? In this case, can you find a nontrivial subgroup?

 

[kathrynag]

G is cyclic means G=<a>, where a is a generator.

 

[Tedjn]

Yes, but reason further. If G is cyclic and its number of elements is composite... is there a nontrivial subgroup?

 

[kathrynag]

I know a cyclic group of order n has exactly one subgroup of order m for each positive divisor m of n.

 

[Office_Shredder]

So can you prove that if G is cyclic with only trivial subgroups, then G is isomorphic to some Z_p with p prime?

The next step after that is to prove that every group has a cyclic subgroup