I am working on a question from Fraleigh's "A First Course In Abstract Algebra":
A torsion group is a group all of whose elements have finite order. A group is torsion free if the identity element is the only element of finite order. A student is asked to prove that if $G$ is a torision group, then so is $G/H$ for every normal subgroup $H$ of $G$. The student writes:
We must show that each element of $G/H$ is of finite order. Let $x \in G/H$
- Why does the instructor reading this proof expect to find nonsense from here on in the students proof?
- What should the student have written?
- Complete the proof.
So I started thinking and just thought of point 3:
We must show that each element of $G/H$ is of finite order.
Let $x \in G/H$.
Observe that $x \in G$ as $G/H \leq G$, but since $G$ is a torsion group, and $x$ is in $G$, $x$ must have have finite order. Q.E.D.
This seems fine to me, but I think I am doing something silly since, the question leads me to believe so.