Determining order of elements and number of automorphisms in dihedral groups

Question

Reviewing some stuff and found myself confused at a few things involving dihedral groups and automorphisms, would very much appreciate some assistance in understanding.

Namely beginning with this, I know $D_4 = \{R_0, R_{90}, R_{180}, R_{270}, H, V, V', D\}.$ By drawing out a square ($D_4$ represents the group of symmetries of a square), and tracing where each of the rotations are brought, I'm able to determine the orders.

But is there a more systematic way to figure this out? I understand the rotations are $a, a^2, a^3, a^4$ and reflections $b, ba, ba^2, ba^3$ but how would one go about determining the order just from knowing this? I can't see a way to figure out the orders of $a, a^2, a^3, a^4$ (reflections are easy since they're all order 2), just by looking at it.

The second one has me a bit confused. So I understand automorphisms preserve order from $f(a)$ and $a \in G$. But why the distinction between the possibilities for $f(a)$ and $f(b)$? If $a$ are the rotations in $D_4$ and $b$ the reflections, then why is $f(b) = a^2$ given as a possibility. $a^2$ is a rotation.

Answer 1

Let me help you systemize (a) using just the information in the description of $D_4$ given in the line above (a). Such a description is called a "presentation."

The orders of $\{ a, a^2, a^3\}$ are derived by computing their powers and applying $a^4=e$.
The order of $b$ is given as $2$. For the others just square them applying $bab=a^3$ along with the other identities.
Thus $(ba)^2=baba=a^3a=e$,
$(ba^2)^2=baabaa=baababba=baaa^3ba=baba=e$, and
$(ba^3)^2=ba^3ba^3=bbabba^3=aa^3=e$.
(Two side comments: 1) You could use these methods to create a multiplication table for this group. 2) The above is not the only way to do these calculations. )

Now let's move on to (c), again relying just on the presentation (description). Your approach is generally correct, so I will just make a few comments.

1. $f(b)$ can't equal $a^2$ because $a^2$ has a square root and since $b$ doesn't have one $f(b)$ can't have one either. An alternative argument: $a^2$ is in the center of $D_4$ but $b$ is not, so $f(b)$ cannot $=a^2$.
2. You need to show that $f(b)f(a)f(b)=f(a)^3$ for all 8 choices of $(f(a),f(b))$.
3. You have to show that $f$ is a bijection. Since we are dealing with a finite group, this can be done by showing that $f$ is either surjective or injective. One way to do this is to show that $x\ne e \implies f(x) \ne e,$ which usually arises naturally while you are doing something else.