The Unapologetic Mathematician

Mathematics for the interested outsider

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 as, 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 - Posted by | Abelian Groups, Algebra, Group theory, Structure of Groups

16 Comments »

  1. I’m not sure that I’m understanding free groups yet.

    Comment by andy | February 8, 2007 | Reply

    • A group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st=smm^{-1}t). Apart from the existence of inverses no other relation exists between the generators of a free group.——wikipedia

      Comment by Luqing Ye | November 16, 2011 | Reply

    • A group is called a free group if no relation exists between its group generators other than the relationship between an element and its inverse required as one of the defining properties of a group. ——wolfram mathworld

      Comment by Luqing Ye | November 16, 2011 | Reply

  2. So are free groups just groups without any composition defined (except canceling with the inverse) ?

    Comment by andy | February 8, 2007 | Reply

  3. Sort of. The elements of a free group are all the finite sequences of generators or their inverses, and words are composed by concatenation of the sequences. The empty sequence is the identity (which you should easily verify), and any word has an inverse (as the post asks you to verify). Does this seem clearer?

    Comment by John Armstrong | February 8, 2007 | Reply

    • Doesn’t the identity have to exist as an element of the group? If so, the elements of a free group consist of all finite sequences of generators or their inverses or the empty sequence, and words are composed by the concatenation of sequences.

      Comment by Doug Spoonwood | May 16, 2011 | Reply

  4. Permutation groups!

    Comment by BadZen | February 8, 2007 | Reply

  5. Oh, you already covered them, cool!

    I was thinking it’d be useful to talk about the actions of x \in G on G, as permutations, and so forth.

    Comment by BadZen | February 8, 2007 | Reply

  6. Indeed it would be useful. Group actions and symmetries are definitely on the agenda. I just want to have homomorphisms under my belt so I can do them neatly, and for that I’ll need functions. Stay tuned.

    Comment by John Armstrong | February 8, 2007 | Reply

  7. […] and Relations Now it’s time for the reason why free groups are so amazingly useful. Let be any set, be the free group on , and be any other group. Now, […]

    Pingback by Generators and Relations « The Unapologetic Mathematician | February 28, 2007 | Reply

  8. […] If we have a group , it may or may not be abelian. We can measure how nonabelian it is with the commutator subgroup . This is the subgroup generated […]

    Pingback by Commutator Subgroups « The Unapologetic Mathematician | March 14, 2007 | Reply

  9. “What should the free Abelian group on n letters look like?”

    All I come up with nothing, so I’m missing something.
    A free group of n letters: a b c d ….. n

    Since they need to be different I dont see how:

    aab * cdc = aabcdc

    cdc * aab = cdcaab

    can be asssociative.

    Comment by Michael D. Cassidy | August 16, 2007 | Reply

  10. Well, you have to have the free group, so start by making strings of generators, just like the free group.

    Then you have to make the multiplication commute. First see what that means for the product of two generators.

    Comment by John Armstrong | August 16, 2007 | Reply

  11. […] Ab-Categories Now that we’ve done a whole lot about enriched categories in the abstract, let’s look at the very useful special case of categories enriched over — the category of abelian groups. […]

    Pingback by Ab-Categories « The Unapologetic Mathematician | September 14, 2007 | Reply

  12. Largely irrelevant, as this post is almost a year old, but there’s a latex typo about halfway down (missing a crowning dollar sign).

    Comment by Jon | December 19, 2007 | Reply

  13. […] must satisfy. The fact that this works out is deeply wrapped up in the universal properties of free groups, but if that sounds scary you don’t have to worry about […]

    Pingback by Some Sample Representations « The Unapologetic Mathematician | September 13, 2010 | Reply


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