Prove that Q has an automorphism of order 3.

Question

Let $$A=\begin{pmatrix}i & 0\\ 0 & -i\end{pmatrix}$$ and

$$B=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}.$$

Let $Q=\langle A,B\rangle.$

Prove that Q has an automorphism of order 3.

Answer 1

As Praphulla Koushik has noted in another answer, the key is to recognize this as the quaternion group: ignoring the usual convention that $I$ denotes the identity matrix and instead calling the identity '$\mathbb{1}$', and changing our variable names from $A$ and $B$ to the suggestive $I$ and $J$, we have $I^2 = \begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix} = -\mathbb{1}$, where of course $(-\mathbb{1})^2=\mathbb{1}$; likewise $J^2 = -\mathbb{1}$. And if we define the matrix $K$ by $K=IJ=\begin{pmatrix}0 & i\\ i & 0\end{pmatrix}$, then we also have $K^2=-\mathbb{1}$; in other words, the three matrices $I, J, K$ satisfy the relations $I^2=J^2=K^2=IJK=-\mathbb{1}$. (Note that the last one derives trivially by expanding one instance of $K=IJ$ in $K^2=-\mathbb{1}$). Furthermore, we have $JKI=-\mathbb{1}$ and $KIJ=-\mathbb{1}$ (you can prove the last two by algebraic manipulations using the relations you already have, without doing any matrix arithmetic; this is a good exercise). A complete list of all the elements in the matrix group would be $\mathbb{1}, -\mathbb{1}, I, J, K, -I, -J, -K$ (where e.g. $-I$ is $-\mathbb{1}\cdot I$) — prove this!

Now, you should see a certain symmetry in the relations among $I, J, K$ that were written above; you should be able to exploit this symmetry to find a morphism $\eta$ by picking suitable 'target' matrices for $\eta(I)$, $\eta(J)$ and $\eta(K)$ and then showing that all of the relations are invariant under application of the morphism $\eta$.

Answer 2

first of all i am sorry for making you to do so much non sense.

First of all i believe that you could recognize your $Q$ to be $Q_8$.

So, you have $Q\cong Q_8=\{\pm 1, \pm i, \pm j, \pm ij\}$.

Now, you need to see if there is a possibility for an automoprphism of $Q_8$ to have order $3$

By an "automorphism of a group" you mean a "homomorphism which is bijective"

So, your required map $\eta : Q\rightarrow Q$ has to be a homomorphism.

For that you need $\eta(a.b)=\eta(a).\eta(b) \forall a,b\in Q$

$\eta(a)=\eta(a.1)=\eta(a).\eta(1)$.Thus, $\eta(1)=1$.

Now, you have $1=\eta(1)=\eta(-1.-1)=\eta(-1).\eta(-1)$ i.e., $(\eta(-1))^2=1$

So, only possible choice of $\eta(-1)$ for non trivial bijective function $\eta$ is $-1$

i.e., $\eta(-1)=-1$

You have $Q=\{\pm1,\pm i,\pm j,\pm ij\}$

Now, let us go for next generator that is $i\in Q_8$

you have six possibilities for $i$ namely $\{\pm i, \pm j, \pm k\}$

you want your map to be of order $3$ so you should not send $i$ to $i$.

with previous messed up post you could be able to see that i can not send $i$ to $-i$

Let us consider a map

$\eta : Q\rightarrow Q $ which send $i\rightarrow j$

please see that this should automatically fix image of $i$ i.e.,

$\eta(-i)=\eta(i)=-j$

Now let us try(cry) for possible image of $j$

For similar reason as in previous messed up (and also for the reason that we need it to be bijective),

I can not send $j$ to $j$ or even $-j$

So, possible choices for image of $j$ are $\{\pm i, \pm ij\}$

Suppose $j$ goes to $i$ Then, we would have

$$i \rightarrow j\\-i\rightarrow -j\\ j\rightarrow i (\text{just now defined})\\-j\rightarrow -i(\text{as $j$ is fixed, so is $-j$})$$

It is upto you to see where does $ij$ and $-ij$ goes to,

just use that $\eta$ is a homomorphism i.e., if $a$ and $b$ are fixed in domain of $\eta$, then $\eta(ab)$ is fixed as $\eta(ab)=\eta(a)\eta(b)$

So, you should be able to see that

$$i \rightarrow j\\-i\rightarrow -j\\ j\rightarrow i \\-j\rightarrow -i\\ij\rightarrow ji=-ij \\ ji\rightarrow -ji=ij$$

For similar reason why $i$ should not be sent to $-i$ you should be able to see $ij$ should not be sent to $-ji$. thus above map is not what we need.

So, next possible choice is $j\rightarrow -i$. I would request you to check yourself that this would then be of order $4$

So, next possibility for image of $j$ is $ij$.

Now, you should be able to see where does other elements should be mapped to namely,

$$ i \rightarrow j\\-i\rightarrow -j\\ j\rightarrow ij \\-j\rightarrow -ij \\ij\rightarrow jij=i \\-ij \rightarrow -i$$

Now, you do not have to check this is a homomorphism because it is defined keeping in mind of homomorphism condition.

You can see clearly that this is a bijection.

So, this is an automorphism.

Coming to the order, we see that :

$$i \rightarrow j\rightarrow ij\rightarrow i\\ -i \rightarrow -j\rightarrow -ij\rightarrow -i\\ j\rightarrow ij\rightarrow i \rightarrow j\\ij\rightarrow i \rightarrow j\rightarrow ij$$

Please check other images to make sure it is of order $3$.

I hope this should work.