Multiplication/Cayley tables for the Dihedral Groups

Question

I am currently doing a group theory problem, which asks for the multiplication table of the dihedral group $D_4$. Having looked up the answer online, I do not understand how some of the elements arose. For example, the elements on the columns and rows are $e,a,a^2,a^3,b,ba,ba^2,ba^3$. I understand how those elements came about, but I don't understand why e.g. $ab,a^2b,a^3b$ are not considered here. Can anyone help?

Answer 1

What you are doing, implicitly, is expressing the group $D_4$ with a presentation. A presentation of a group is made of two parts: a set of generators, and a set of relations.

Generators are commonly denoted with lowercase letters: $a,b,c,...$

Relations are expressions that generators satisfy: for instance, normally you have some relations with a single generator $a^4=e$, $b^2=e$, etc. , and then relations that tell you how the generators combine, such as $ab=ba$ (commutativity), or more complex ones like $ab=cba$.

Given a presentation, it is not easy in general to understand what group you are looking at: this is because the group is defined as all words in the generators (think $abba$) modulo the relations (meaning that if $ab=ba$, $abba=abab=baba$ etc.). So... you can have words that look different, but are actually the same when you apply the relations.

In your case, $D_4$ is given by $$\langle a,b \; |\; a^4=b^2=e, ab=ba^{-1} \text{ (or } b^{-1}ab=a^{-1})\rangle $$

So, for example:

$$ab = ba^{-1} = ba^3$$ $$a^2b = a(ba^{-1}) = ba^{-1}a^{-1} = ba^2$$

and, in general, whatever expression you consider can be reduced to one of the eight elements given by your book.