1. ## factor group question

Let G be an Abelian group and let H be the subgroup consisting of all elements of G that have finite order. Prove that every non-identity element in G/H has infinite order.
No clue were to go, or even start from. Any help would be appreciated.

2. Originally Posted by wutang
Let G be an Abelian group and let H be the subgroup consisting of all elements of G that have finite order. Prove that every non-identity element in G/H has infinite order.
No clue were to go, or even start from. Any help would be appreciated.

$\displaystyle gH\in G\slash H$ has finite order $\displaystyle \Longrightarrow g^nH=H\Longrightarrow g^n\in H$ , for some $\displaystyle n\in\mathbb{N}\Longrightarrow g$ has finite order and thus...

Tonio

3. so I assume that their exists an element gH in G/H that has finite order to get a contradiction? I still do not understand it fully.

4. Originally Posted by wutang
so I assume that their exists an element gH in G/H that has finite order to get a contradiction? I still do not understand it fully.

What I wrote actually means: if an element has finite order in $\displaystyle G\slash H$ then that element must be the unit element, and thus...well, and thus exactly what

you want to prove, right?

Tonio