MaBloWriMo 10: Groups

So what is a group? Intuitively, a group consists of a set of things $G$ , such that

there is a way to combine any two things together
there is a special thing which has no effect when combined with other things
it is always possible to undo the combining by doing yet more combining.

“Stuff that you can combine” is really common. For example, think of integers (you can add two integers to get another number), or libraries (you can combine two libraries into a bigger library), or train tracks (you can lay two train tracks end-to-end to get a longer train track), or functions (you can compose two functions to get another function), and so on. In each of these examples there is also a special thing with no effect: the integer zero; a library with no books; a train track of length zero (though you might argue with me about whether you can really have a “train track of length zero”…); and the identity function $f(x) = x$ .

However, being able to undo the combining is a strong requirement, and in many cases it is not possible. For example, once you have combined two libraries, it is not possible to get back one of the original libraries just by combining more libraries. Likewise, you can only make train tracks longer by combining, so there’s no way to undo the combining. (In general, you can’t undo function composition either; I’ll let you think about why.) The example with integers, on the other hand, does allow undoing: for example, if we start with 3 and add 4 to it, we can get back to 3 by adding yet again: $3 + 4 + (-4) = 3$ . The $4$ and $-4$ “cancel each other out”. Of course, this is exactly the idea of negative numbers.

Enough with the handwavy intuition for now; here’s the formal definition of a group. A group consists of

a set $G$
a special element $1 \in G$
a binary operation $\cdot$ on $G$

such that

$\cdot$ is associative, that is, $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ whenever $a$ , $b$ , and $c$ are elements of $G$
$1$ is the identity for $\cdot$ , that is, $1 \cdot a = a \cdot 1 = a$ for every $a \in G$
For every element $a \in G$ , there is another element $a^{-1} \in G$ , called the “inverse” of $a$ , such that $a \cdot a^{-1} = a^{-1} \cdot a = 1$ .