The Unapologetic Mathematician

A few more groups

I want to throw out a few more examples of groups before I move deeper into the theory.

First up: Abelian groups. These are more a general class of groups than an example like permutation groups were. They are distinguished by the fact that the composition is “commutative” — it doesn’t matter what order the group elements come in. The composition $ab$ is the same as the composition $ba$.

All the groups I’ve mentioned so far, except for permutations and rotations, are Abelian. It’s common when dealing with an abelian group to write the composition as “+”, the identity as “0″, and the inverse of $a$ as $-a$. Let’s recap the axioms for an Abelian group in this notation.

1. For any $a$, $b$, and $c$: $(a+b)+c=a+(b+c)$
2. There is an element 0 so that for any $a$: $a+0=a=0+a$
3. For every $a$ there is an element $-a$ so that: $a+(-a)=0=(-a)+a$
4. For any $a$ and $b$: $a+b=b+a$

Abelian groups are really fantastically important. Many later algebraic structures start with an Abelian group and add structure to it, just as a group starts with a set and adds structure to it. We’ll see many examples of these later.

The other thing I want to mention is a free group. As the name might imply, this is a group with absolutely no restrictions other than the group axioms. We start by picking some basic pieces, sometimes called “generators” or “letters”, and then just start writing out whatever “words” the rules of group theory allow.

Let’s start with the free group on one letter: $F_1$. We definitely have the identity element — written “1″ — and we throw in our single letter $a$. We can compose this with itself however many times we like by just writing letters next to each other: $aa$, $aaa$, $aaaa$, and so on. We also need an inverse, $a^{-1}$. We can use $a$ and $a^{-1}$ to build up long words like $aaa^{-1}aaa^{-1}aaaa^{-1}a^{-1}aaaaa^{-1}aaaa$. But notice that whenever an $a$ and an $a^{-1}$ sit next to each other they cancel. That collapses this long word down to $aaaaaaaaa$. We see that in $F_1$ all words look like $a^n$, where a positive $n$ means a string of $n$ $a$s, a negative $n$ means a string of $|n|$ $a^{-1}$s, and $n=0$ for the identity. We compose just by adding the exponents.

The free group on two letters, $F_2$ gets a lot more complicated. We again start with the identity and throw in letters $a$ and $b$. Now we can build up all sorts of words like $aba^{-1}aa^{-1}abba^{-1}b^{-1}aaabb^{-1}baab$. But now we can’t do anything to pull $a$ and $b$ past each other. Letters only cancel their inverses when they’re right next to each other, so this word only collapses to $abbba^{-1}b^{-1}aaabaab$. That’s the best we can do. Free groups on more generators are pretty much the same, but with more basic symbols.

Composition of words $w_1$ and $w_2$ just writes them one after another, cancelling whatever possible in the middle. For example, in $F_3$ let’s say $w_1=abcbac$ and $w_2=c^{-1}a^{-1}bc^{-1}ab$. We write them together (in order!) as $abcbacc^{-1}a^{-1}bc^{-1}ab$ and cancel inverses where we can to get $w_1w_2=abcbbc^{-1}ab$.

Free groups seem hideously complicated at first, but they aren’t so bad once you get used to them. They’re also tremendously useful, as we’ll soon see. They’re the primordial groups, with absolutely nothing extra beyond the bare minimum of what’s needed to make a group.

Some points to ponder

• What is the inverse of a word in a free group?
• What should the free Abelian group on $n$ letters look like?

February 6, 2007