If G is an abelian group and n>1 an integer, let A_n = { a^n, a is an element of G}. Prove that A_n is a subgroup of G
We use the one step subgroup test.
First note that $\displaystyle e=e^n$, so $\displaystyle A_n\neq\emptyset$.
Now, take $\displaystyle a^n,b^n\in A_n$. It follows then
$\displaystyle a^n\left(b^n\right)^{-1}=a^n\left(b^{-1}\right)^n=\left(ab^{-1}\right)^n$.
Since $\displaystyle ab^{-1}$ is in the right form, it follows that $\displaystyle A_n\leq G$
Does this make sense?
No not really.
What I tried doing was showing that A_n is closed, associative, has an identity element and an inverse element.
My problem was I got an identity element e = a^0
and inverse = a^-n.. I suppose both of them are wrng because th question says n>1....what do you think I did wrong..
Thnx
To prove something is a subgroup you merely need to show Closure and that Inverses exist. You get associativity for free because you are in a group (think about it...), and the identity because inverses exist (if $\displaystyle a$ and $\displaystyle a^{-1} \in H$ and $\displaystyle aa^{-1} \in H$ then as $\displaystyle aa^{-1}=1$ we have that $\displaystyle 1 \in H$).
Firstly, note that your identity is $\displaystyle e^0$. Secondly, does the question mention "finite" anywhere?
No the question does not mention infinite..
Alo bare with me but am still confused as to why the identity element is e^0
I though a * e = a
and in this case since A_n = a^n
so is a ^n * e = a^n
that is how I got e = a^0
what am doing wrong
Thanks for your help so far
The element $\displaystyle a$ is any element of the group, while the $\displaystyle n$ remains fixed.. For instance, take $\displaystyle G=C_8=\{1,g,g^2, \ldots, g^7\}$, with $\displaystyle g^8=1$. Let us look at this group when $\displaystyle n=2$, $\displaystyle A_2$. This is the set of squares of all elements of the group:
$\displaystyle A_2 = \{1^2=1, g^2, g^4, g^6 \}$.
The identity element is NOT $\displaystyle a^0$ - the $\displaystyle n$ is fixed. It is the elements of the group that we change. The identity occurs when $\displaystyle a=1$ as $\displaystyle 1^n=1 \forall n \in \mathbb{N}$.
If $\displaystyle g \in G$ then $\displaystyle g^{-1} \in G$. Thus $\displaystyle g^n \in A_n$ and $\displaystyle (g^{-1})^n = g^{-n} \in A_n$.
For closure you need the abelian condition of $\displaystyle G$.