Alternating group problem in abstract algebra 
I understand the great majority of what the problem is asking. It asks:
We have two different permutations of length n and we want to show that the inverse of the inverse of the permutation of the permutation of these two different permutations are in the alternating group length n.
I'm having problems understanding this alternating group...I only know basic group theory and I'm not exactly sure what this means.
Here's an example of my understanding of the problem:
Let $ \begin{matrix} 1&2&3 \\ 2&3&1 \end{matrix}$ denote $\alpha$ and $\begin{matrix} 1&2&3 \\ 3&1&2 \end{matrix}$ denote $\beta$. 
Then $\begin{matrix} 2&3&1 \\ 1&2&3 \end{matrix}$ denotes $ \alpha^{-1}$ and $ \begin{matrix} 3&1&2 \\ 1&2&3 \end{matrix}$ denotes $\beta^{-1}$.
Therefore $ \alpha^{-1} \circ \beta^{-1} \circ \alpha \circ \beta = 1 \ 2 \ 3$
And this is supposed to be in this $A_n$. But I'm not entirely sure why or what that means.
 A: Background reminder:
A transposition is a permutation that simply switches two entries, i.e. $\tau\in S_n$ is a transposition interchanging $i$ and $j$ (for $i\not=j$) if $\tau(i)=j$ and $\tau(j)=i$ and $\tau(k)=k$ for $k\not=i,j$.
Every permutation $\alpha\in S_n$ can be written as a product of transpositions.
Now one defines $\mathrm{sign}\,\alpha=(-1)^n$ if $\alpha$ can be written as a product of $n$ transpositions. This is a well-defined (!) group homomorphism $S_n\rightarrow \{\pm 1\}$.
$A_n$ can be defined as the set of all even permutations, i.e.
$$A_n=\{\alpha\in S_n\,:\,\mathrm{sign}(\alpha)=1\}$$
Your problem:
We have to show that $\gamma=\alpha^{-1}\circ\beta^{-1}\circ\alpha\circ\beta$ can be written as a composition of an even number of transpositions.
We know that $\alpha$ and $\beta$ can be written as products of transpositions, say $n$ and $m$ many, respectively. Then $\alpha^{-1}$ and $\beta^{-1}$ are also products of $n$, respectively $m$ transpositions (why?).
So we can write $\gamma$ as a product of $n+m+n+m=2(n+m)$ transpositions, i.e. $\gamma\in A_n$.
