Defining Dihedral groups using reflections.

Question

I'm trying to understand Dihedral groups $D_n$ of order $2n$. I'm a beginner at Group theory and my textbook uses Dihedral groups to motivate Group Theory so I haven't really began studyng group theory yet and hence I do not understand the "relators" or "free groups" or "presentation" yet so if you could use simple English to answer my question, I'd appreciate it.

I often (frankly always) see dihedral groups being defined using rotational and reflection symmetries like so:

$r$ is rotation of $n-$gon by $2π/n$ and $f$ is reflection about a fixed axis.

So, $D_n = \{r^if^j \left(\text{ or } f^j r^i \right) : 1≤i≤n, 1≤j≤2\}$

My question is, is it possible to define $D_n$ without using rotational symmetries, at all?

If you multiply two reflections you end up with a rotation. So is it possible to define $D_n$ by using $f_1,f_2, \ldots, f_n$ being reflections about each of the $n$ axes then $$D_n= \{f_i^a f_j^b: 1≤a,b≤2, 1≤i,j≤n \}$$

Two interesting questions arise here:

1. Why use rotational symmetries at all? Multiplying reflections produces rotations but two rotations do not produce a reflection. Isn't the second definition better or are there shortcomings to it?

2. Does this change the number of generators? In the first definition, $D_n$ is generated by two guys: $r$ and $f$ while in the second definition, you have $n$ generators?

Answer 1

It is important to remember that a group is not just a set, it is a set endowed with a particular algebraic structure. To put it simply: To define a group it does not suffice to list the elements of the underlying set, you also have to define an operation on that set satisfying the group axioms. The statements

$$D_n = \{f^i r^j | 1 \leq i \leq 2, 1 \leq j \leq n \} $$

does not make sense on its own because it does not tell you how all these elements multiply. When you say that multiplication is just composition, you assume that you know what composition means and that forces you to go back to thinking about reflections and flips of an $n$-gon. You can avoid this by giving what is known as a group presentation. In a group presentation, you give a number of generators, their orders and a number of relations that uniquely determine the multiplication of all elements. In the case of $D_n$ such a presentation is

$$ \langle r, f | r^n = e, f^2 = e, frf^{-1} = r^{-1} \rangle$$