G is for Group

I have said and written that the group is arguably the most useful mathematical concept that most people have never heard of.  The following definition might seem technical, but I assure you that it is really nothing more than formalizing the “symmetry” concept.

A group is a set G of objects (maybe a finite set, maybe infinite, for now I don’t care), together with a binary operation \ast.  In other words, if x and y are objects in the group, then x\ast y is an object in the group.  (If you’re into technical lingo, \ast is like a function that takes two arguments; addition, subtraction, and multiplication are all binary operations on numbers, for example.)  But not just any old binary operation gives a group: it must satisfy the following three properties.

  1. associativity: x\ast (y\ast z) = (x\ast y) \ast z.
  2. identity: there is a special object in the group, \mathrm{id}_G, such that x\ast \mathrm{id}_G = \mathrm{id}_G\ast x.
  3. inverses: for every object x, there is an inverse object x^{-1} such that x\ast x^{-1}=x^{-1}\ast x=\mathrm{id}_G.

The objects in a group might be just about anything: numbers of one sort or another, matrices, functions, symmetries of an object, abstract symbols, etc. The heart of the group is not in how we name its elements, but rather in the binary operation.

Examples of groups are legion.  The permutations of any set form a group. The real numbers form a group under addition.  Meanwhile, the nonzero real numbers form a group under multiplication.  The symmetries of a square form a group under composition.  Square matrices of any particular size form a group under multiplication, as long as you only include matrices with inverses.

The operation is often called multiplication and written with the same kind of notation.  I used a star in the definition, but more common is to use no symbol at all, just as we do for multiplication in Mrs. Gunderson’s algebra class (i.e. xyz for x\ast y \ast z).  Likewise we write x^2 for xx, and so on.

Sometimes people think of the operation additively.  We won’t do this here, and the reason why we won’t deserves attention.  Note that while we are assuming that our operation is associative, like addition and multiplication, we are not assuming that it is commutative.  That is, order can matter: x\ast y need not be the same as y\ast x.  It’s like taking a shower and getting dressed—if you do them in the other order, you probably don’t get the desired results.  If x\ast y=y\ast x, if order doesn’t matter, then your group is especially simple, it’s called a commutative, or abelian, group.  For psychological reasons, mathematicians never (as far as I know) use addition-language for binary operations that don’t commute.  If you think about the integers with addition as your prototypical example of a group, then you might think it’s a good idea to only think about commutative groups.  In contexts when people only care about commutative groups, addition notation should be used.  But in some sense this is the wrong perspective for us to go in with.

Actually multiplication isn’t really the best term to use either.  You should really be thinking of the operation as composition of functions.  If, as I have strongly implied, groups represent symmetry structure, then the elements can be viewed as symmetries, i.e. as transformations.  Of course, a certain amount of conflation of the ideas of multiplication and composition is pervasive in mathematical lingo because of matrix theory.  You probably learned something about matrices in high school, in particular that they have a multiplication operation which is not commutative; it turns out that matrices represent certain kinds of functions on linear spaces (whatever those are), and multiplication of matrices corresponds to composition of the functions.

The core idea of the group as a group of transformations under composition explains why we want groups to be associative, but not necessarily commutative.  What about the other two properties?  They say that there should be an element of every group which “does nothing”, which behaves the way 0 does for addition and 1 does for multiplication, and that every element should have an “inverse”.  These are restatement of the convention that the transformation which does nothing is a symmetry and the idea that symmetries are “undo-able”.    You’d never think of smashing something to bits as a symmetry, because it is not readily undone.  Symmetries leave an object “structurally the same as it was”, so there will always be another (possibly the same) symmetry to undo any given symmetry.

Technical mathematical definitions are almost always an attempt to formulate and formalize a very primitive idea, it’s just a matter of seeing it.

(This is more motivation for counting the trivial transformation, the one that doesn’t change anything, as a symmetry.  If we excluded the trivial symmetry we’d get the undesirable side effect that the composition of two symmetries would not necessarily be a symmetry.  Group theory would be much more confusing and complicated if we artificially excluded from each group the identity element.)

Let’s look at a couple examples. Recall the two colored squares from last time.

Two Colored SquaresWe’ve already said that the symmetry group of the left square has four elements: identity, quarter-turn, half-turn, and three-quarter turn.  We can write these as 1, r, r^2, r^3.  Then the multiplication looks just like we’d expect (r^2 r = r^3, say), with the understanding that r^4=1, r^5=r, etc.

The symmetry group of the right square is different.  It’s four symmetries are: identity i, horizontal flip h, vertical flip v, and half-turn t.  The multiplication table here is very different.  it=ti=t, iv=vi=v, ih=hi=h, of course.  Also h^2=v^2=t^2=i (doing any symmetry twice brings us back where we started), and the product of any two of h,v,t is the third one.

We see that our prior observations about symmetry fit naturally into this language and notation.

And symmetries are not just geometric.  Here’s a frivolous example.  Consider the following eight permutations of the four symbols \heartsuit, \diamondsuit, \clubsuit, \spadesuit.

  1. \heartsuit\to\heartsuit, \diamondsuit\to\diamondsuit, \clubsuit\to\clubsuit, \spadesuit\to\spadesuit
  2. \heartsuit\to\diamondsuit, \diamondsuit\to\heartsuit, \clubsuit\to\clubsuit, \spadesuit\to\spadesuit
  3. \heartsuit\to\heartsuit, \diamondsuit\to\diamondsuit, \clubsuit\to\spadesuit, \spadesuit\to\clubsuit
  4. \heartsuit\to\diamondsuit, \diamondsuit\to\heartsuit, \clubsuit\to\spadesuit, \spadesuit\to\clubsuit
  5. \heartsuit\to\spadesuit, \diamondsuit\to\clubsuit, \clubsuit\to\diamondsuit, \spadesuit\to\heartsuit
  6. \heartsuit\to\clubsuit, \diamondsuit\to\spadesuit, \clubsuit\to\diamondsuit, \spadesuit\to\heartsuit
  7. \heartsuit\to\spadesuit, \diamondsuit\to\clubsuit, \clubsuit\to\heartsuit, \spadesuit\to\diamondsuit
  8. \heartsuit\to\clubsuit, \diamondsuit\to\spadesuit, \clubsuit\to\heartsuit, \spadesuit\to\diamondsuit

You can check this is a group (under composition), if you’re so inclined, but why is it interesting?  Is it a symmetry group in some sense?  Note that these are precisely the permutations of the suits that preserve “same-color-ness”.  In some sense, this is the symmetry group of a deck of playing cards preserving which cards are and are not the same color.  So, if a group theorist plays a lot of Klondike solitaire, she might care about this group.

Other (less frivolous) structures, such as number systems, can have symmetry groups.  (Now we’re getting somewhere.) The content of the first alien post is that the symmetry group of \mathbb{C} relative to \mathbb{R} has two elements, the trivial one and complex conjugation, and that \pm i are related by the group.  The second alien post leads us to consider the symmetry group of the algebraic number system relative to \mathbb{Q}.  I didn’t give a nice description of this symmetry group because I don’t understand it—nobody does.  (Indeed, trying to understand this group is a motivation for much of modern number theory, including most of what I do.)  But we can understand some simple pieces, as suggested in that post.  The symmetry group of the rationals, together with $latex\sqrt 2$, relative to the rationals is another two-element group.  The symmetry group of the rationals, together with $latex\sqrt 3$, relative to the rationals is another (isomorphic) two-element group.  Likewise if for \sqrt 6.  But these three groups interact.  If we look at the symmetry group of the rationals with \sqrt 2, \sqrt3 \sqrt 6 relative to the rationals, combining all these two-element groups, then we only get four symmetries, not the eight you might have expected.

(This just in from the getting-ahead-of-ourselves department.  The four symmetries are the trivial one, the one that negates \sqrt2 and \sqrt 3, the one that negates \sqrt2 and \sqrt 6, and the one that negates \sqrt 3 and \sqrt 6.  This is isomorphic to the symmetry group of the rectangle.)

In some sense, this is Galois theory (the purpose of this arc, remember); one description of Galois theory is the study of symmetry groups of number systems.  The next step in our journey will be to make better sense of what “number system” means in this context.

(If this is the first time you’ve heard about groups, a good next source would be the wikipedia entry, which has some good pictures and gives a good feel for how the theory goes; as usual for a mathematical wikipedia entry, though, it also includes some material which is much more high-level than the rest, so don’t lose any sleep over phrases like “étale fundamental group”.)

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