Show that ##G\simeq \mathbb{Z}/2p\mathbb{Z}##

[JojoF]

Homework Statement

Let $G$ be a group of order $2p$ with p a prime and odd number.

a) We suppose $G$ as abelian. Show that $G \simeq \mathbb{Z}/2p\mathbb{Z}$

Homework Equations

The Attempt at a Solution

Intuitively I see why but I would like some suggestion of what trajectory I could take to prove this.

I proved in an earlier problem that all groups with a prime order is a cyclic group.
I am sure it is a Sylow theorems problem.

Thanks!

 

[member 587159]

Homework Statement

Let $G$ be a group of order $2p$ with p a prime and odd number.

a) We suppose $G$ as abelian. Show that $G \simeq \mathbb{Z}/2p\mathbb{Z}$

Homework Equations

The Attempt at a Solution

Intuitively I see why but I would like some suggestion of what trajectory I could take to prove this.

I proved in an earlier problem that all groups with a prime order is a cyclic group.
I am sure it is a Sylow theorems problem.

Thanks!

A (possible) hint: show that $G$ is cyclic and the result will follow because any two cyclic groups of the same order are isomorphic.

 

[fresh_42]

The assumption, that $G$ is Abelian is crucial, as the example with $p=3$ and $Sym(3)$ shows, so this has to play a role in your proof. E.g. all subgroups are automatically normal. I would concentrate on the first Sylow theorem.