Question of lagrange theorem converse.

[betty2301]

Homework Statement

Let G be an abelian group. Suppose p divides ord(G) where p is prime no. Prove G has a subgroup of order p.

Homework Equations

lagrange theorem converse

The Attempt at a Solution

i know the converse is lagrange theorem and easy and this is not the case.
I know abelian has something to do with this and prime no is also something special. I also believe the subgroup of order p is cyclic?
thx

 

[jbunniii]

Yes, any group of prime order is cyclic.

The theorem you are being asked to prove is called Cauchy's theorem, and it is true even if G is not abelian. There are numerous ways to prove it, depending on what results you already know. Maybe the most straightforward way is induction on $|G|$.