1)
Show that the set of 2×2 matrices with non zero determinat together with multiplication form a group $(H,*)$ where
$$H=\left.\left\{\begin{pmatrix} a&b\\c&d\end{pmatrix}\right|\; a,b,c,d\in \mathbb{Z}_2, ad-bc\neq 0 \right\}$$
Show that this group is isomorphic to $S_{3}$
2) identify elements of order 2 and show they are characterized by their traces
Hint show that the group acts naturally on the vector space $\mathbb{Z_{2}}\oplus \mathbb{Z_{2}}$ and permutes the non-zero elements.
part 1 is very simple. I showed that the multiplications is an operation on this set, the associativity is inherited from the 2×2 matrices, the id element is just the id matrix and the set is closed under taking inverses.
How to do the part 2)? How to create such isomorphism? What does it mean that the elements of order 2 are characterized by their traces???
Best Answer
Hint: Let $H$ act canonically on the set of non-zero vectors of $\Bbb Z_2\oplus \Bbb Z_2$ and show that this action is faithful. Then $H$ must be isomorphic to a subgroup of $S_3$ and simply by counting it must be all of $S_3$.
"The elements of order $2$ are characterised by their trace" means that there is a condition of the form "$A\in H$ has order $2$ if and only if $\text{trace}A=\;...$
Edit: Think of the non-zero vectors $\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}$ and $\begin{pmatrix}1\\1\end{pmatrix}$ as the numbers $1, 2, 3$. Formally, we take a bijection (a "renaming" if you will) $f$ from the set of non-zero vectors into the set $\{1,2,3\}$. Now, regular matrices map non-zero vectors onto non-zero vectors, so we can define a map $\varphi:GL(2,\Bbb Z_2)\to S_3$ by the rule $A\mapsto \sigma$, where $\sigma(i)=f(Af^{-1}(i))$. That means that if, for example, $A\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}0\\1\end{pmatrix}$, then the corresponding $\sigma$ will satisfy $\sigma(1)=2$. This $\varphi$ will give you the desired isomorphism.