Group Multiplication Table
Fun fact to know: in 1992 Ales Drápal proved that if two finite groups agree on 89% on their multiplication tables, the groups must be isomorphic! He conjectured that the same holds true if the tables agree on 75% of their entries. The conjecture has not been proved yet. See also Groups St. Andrews 2001 at Oxford, featuring the paper of Drápal On the distance of 2-groups and 3-groups.
"Completely determine" means two things simultaneously. Firstly, "determine" here means that each entry in the multiplication table can be filled in by one of those six elements, which is basically what you said. Secondly, "completely" means that there is no ambiguity: some entries in the multiplication table can be worked out in different ways, and no matter how many different ways you try to work some given entry out, you will get the same thing.
It's entirely conceivable that you might be faced with an equation like $ab^{-1} = cd^{-1}$. No matter how hard you try to get rid of the inverses, you can't.
The author knew because this is a very standard group called $D_6$ (sometimes called $D_3$). You'll know it too once you meet the dihedral groups. But that's not a good answer. Another way to motivate this: notice that every element that he claims lives in that six-element set is of the form $a^ib^j$, where $i$ is 0 or 1 and $j$ is 0, 1 or 2. But what about $ba$, or other things that don't fit into this form? Our group axioms tell us that, since we have an element $b$ and an element $a$, we should be able to multiply them. So we need to know what to do when we have something of the 'wrong' form. And these wrong forms can be broken up into three types:
- $a^i$ where $i > 1$ or $i < 0$,
- $b^j$ where $j > 2$ or $j < 0$,
- $ba$ (multiplied the 'wrong' way round).
These three relations tell us exactly what to do in each of those situations.
Exactly the same reason children write out their times tables. It shows you what's going on! It gives you much more of a clue of the structure of the group than you might expect at this point; on the other hand, you're right to be suspicious of them, because nobody uses them after a certain level. But pedagogically they're very important.
The use of inverses (or more precisely, the use of negative exponents) is superfluous in the case of finite groups. In a finite group, every element has finite order. In other words, for every element $a$, there is a positive integer $k$ such that $a^k=e$. Hence instead of writing $a^{-1}$, one can write $a^{k-1}$, which is the same thing. In your example $b^{-1}=b^2$.
On the other hand, as soon as your group has elements of infinite order, it is no longer possible to write inverses as positive powers of the elements, and so the "$-1$" notation may become necessary. This can only happen in groups of infinite size.
The three relations in your example tell you
- what the order of $a$ is,
- what the order of $b$ is,
- how $a$ and $b$ move past each other.
With two generators, this is enough to determine the results of arbitrarily complicated multiplications, so you won't need additional relations.
In groups of large size, it may be impractical to write out the multiplication table. What's important is being able to produce the multiplication table in principle. The group is only defined once you've specified how to multiply arbitrary elements of the group. The multiplication table is one way to do that, but if you can prove that your relations allow you to compute arbitrary products, that's fine too.