Proving G is Cyclic if No Subgroups Other than G and {e} Exist

[vedicguru]

Homework Statement

Prove that if a group G has no subgroup other than G and {e}, then G is cyclic...

Homework Equations

The Attempt at a Solution

we could say that, let a E G - {e} then we construct <a>...

 

[quasar987]

Sounds good. Why are you having doubts?

 

[vedicguru]

how do I construct <a> ??

 

[HallsofIvy]

Well, you were the one who used that notation- what is the definition of "<a>"?

 

[quasar987]

Huh? Well by definition, <a>={..., a^-1,e,a,a²,a³,...}. Since <a> is a subgroup by construction and since it is not equal to {e}, it is equal to G.

 

[vedicguru]

oh yeah sure! I meant how would say, after constructing it! I forgot it is equal to {e} it is equal to G...this is what I get!
We could say that, let a E G - {e} then we construct <a> =
a, a^2, a^3... a^n=1 <- cyclic subgroup generated by element a.
Since <a> is subgroup, it must coincide with G (because G is the only subgroup), but then a generates G: <a> = G, which is definition of G being cyclic...
are we done after saying this statement or should we have to mention something else?
thanks for offering help!

 

[quasar987]

Well what you wrote for the definition of <a> is incorrect, and don't forget to say "since <a> is not {e}, it is G".

 

[vedicguru]

ohh yeah I am sorry about that...I was in hurry or something to write it down! but yeah I believe everything else is fine and I would not forget to mention that <a> is not equal to {e}..it is G!
Thanks!