# Gracious Living

Groups
November 11, 2010, 19:00
Filed under: Algebra, Math | Tags: , , , , ,

New topic!  To be mixed in with the topology.  We’ve been talking a lot about a strange thing called $\mathbb{R}$ and some understanding of algebra will help us pin down what exactly this is.  (“Algebra” here is not the same as the “algebra” you learn in high school, by the way, though there is a relation.)  I’d also like to bring up group actions in the context of quotient spaces, and maybe start discussing the fundamental group of a topological space.

An introduction: Groups are one of the first things most math students learn in college, and probably one of the more  useful.  Though study of groups themselves is mostly viewed as a closed subject these days, the idea underlies many more current ones, and is the foundation of modern algebra.  The idea was essentially invented by the mathematician Évariste Galois, who used it to study the interaction between the roots and coefficients of polynomials1.  Shortly after, the idea of a continuous group of transformations (of, say, figures in geometry) was studied by people like Felix Klein and Sophus Lie; with discrete groups, problems were usually more combinatorial in nature, and study has historically centered on ways of efficiently categorizing and understanding large groups (such as “representing” them as linear transformations of a vector space, or writing them as constructions made of simpler groups, or as generators and relations, or so on).

The group concept is a highly abstract one, but once you begin to understand it, you see how wide its applications are.  Below the fold, let’s do some math!

A group is a set $G$ with a binary operation, which is just a function $\circ:G\times G\rightarrow G$, usually written as something like $g\circ h$ rather than $\circ(g,h)$.  The binary operation has to satisfy the following three conditions:

• Associativity: $g\circ (h\circ j)=(g\circ h)\circ j$ for all $g,h,j\in G$.
• Identity: There is an element $e\in G$ with $e\circ g=g\circ e=g$ for all $g\in G$.
• Inverses: For every $g\in G$, there’s an element $g^{-1}\in G$ with $g\circ g^{-1}=g^{-1}\circ g=e$.

One thing that should be mentioned right off the bat is that the group operation isn’t required to be commutative: that is, it’s not true in general that $g\circ h=h\circ g$.  If this property holds for all elements of $G$, $G$ is said to be abelian, after the Norwegian mathematican Niels Henrik Abel.  As we’ll see, abelian groups have a different and simpler theory than general groups.

A word about notation: I used $\circ$ above because that’s how I learned it in school, but obviously $+,*,\cdot,\bullet,$ et cetera can be used for a group operation.  Most of the time, you just suppress the operation completely and write $gh$ for the product of $g$ and $h$, and $g^3$ for $ggg$.  Oh, it’s also common to refer to the operation as multiplication despite its possible noncommutativity.  The exception is abelian groups, when we prefer to use $+$ for the operation and call it addition: this is called “writing a group additively.”

Okay, let’s look at a couple examples.  The integers under addition are a group — abelian, even.  The identity is 0, and the inverse of $a$ is $-a$.  The natural numbers under addition aren’t, since most of them don’t have inverses.  Likewise, the integers aren’t a group under multiplication, since we have $3$ but not its inverse $1/3$.  The rationals under multiplication almost solve this problem — but $0$ has no multiplicative inverse, so we have to remove it to make it a group.  The resulting set $\mathbb{Q}-\{0\}$ is often written $\mathbb{Q}^\times$ for this reason.  Similarly, $\mathbb{R}^\times$ and $\mathbb{C}^\times$ are multiplicative groups, and $\mathbb{Q},\mathbb{R}$, and $\mathbb{C}$ are groups under addition too.

How about some non-abelian examples?  Probably one of the motivating examples (and why I used $\circ$ above) is the group of bijections of a set to itself, under composition of functions.  Every set has an identity map that’s a bijection, and serves as the identity of this group; the inverse function is similarly the inverse in the group.  Similarly, a topological space gives you a group of homeomorphisms to itself.  Obviously these groups are non-abelian in general: if $f(x)=2x,g(x)=x+1$ on $\mathbb{R}$, then $f\circ g(x)=2(x+1)=2x+2$, but $g\circ f(x)=2x+1$.

One of the best ways of thinking of groups, though, is symmetry.  Let’s look at something as simple as a square.  I’ve drawn one at left but it might help you to draw your own to play around with.  Clearly, reflecting across any of the green lines gives you the same square again; you can also rotate the square by 90, 180, or 270 degrees clockwise (red arrows).  And you also have the identity function!

What we can do is think of these symmetries as elements of a group.  To get us started, I’ve labeled the clockwise rotation $r$ and the reflection about the vertical line as $f$, for “flip”.  Obviously, $f=f^{-1}$: if we flip and then flip again, we get our original square back.  Likewise, $r^4=e$, a 360 degree rotation, and so $r^3=r^{-1}$.  For more complicated operations, it helps to label the vertices: so our starting square is ABCD, and $f$ takes it to BADC, while $r$ takes it to DABC.  Observe that if we had any other labelling, $f$ and $r$ would do “essentially the same thing”: $f$ switches the first two and last two letters, and $r$ puts the last letter at the beginning.

So, in particular, if we apply $f$, we get BADC; we can then apply $r$ to move the C up front and get CBAD.  Call this move $rf$ (I’m using the same order as for composition of functions, where the thing you do first goes last.)  This is just reflection about one of the diagonal lines!  Work out for yourself — you can probably already guess — what $r^2f$ and $r^3f$ are.  And $r^4f=f$, since $r^4=e$.

Now look at $fr^3$.  Apply $r$ three times to get BCDA, and then flip to CBAD.  This is the same as $rf$!  So we have another equation that holds for this group: $rf=fr^3$.  Then $r^2f=r(rf)=r(fr^3)=(rf)r^3=fr^6=fr^2(r^4)=fr^2$, and similarly, $r^3f=fr$.  We can “carry $r$ across $f$,” but its direction gets reversed.

Okay, so I claim that these equations give you everything you need to know to understand the group.  Because given any product of $r$‘s and $f$‘s, we can just move all the $r$‘s to the right, and then cancel out copies of $r^4$ and $f^2$.  This means that any element of the group can be written in the form $f^ir^j$, where $i=0,1,j=0,1,2,3$ (and $f^0r^0=e$).  In particular, the group only has eight distinct elements!  And we know what they are, too: reflections across the four lines of symmetry, and rotations by 0, 90, 180, or 270 degrees!  As another consequence, if you’ve reflected the square across any line, you can’t rotate it back to where it was — you have to reflect it again at some point, though it doesn’t matter which line you use.

The name for this group is the dihedral group of order 8, or $D_8$ for short.  The order of a group is its cardinality (its number of elements, for finite groups), and “dihedral” groups are the groups of symmetries of regular polygons.  (The word basically means “two-sided,” probably referring to the “front” and “back” of the polygon — “hedral” as in “polyhedron” here.)  If you feel like investigating further, try doing what we did above for $D_6$, the group of symmetries of an equilateral triangle, or for other polygons.  Try to find a pattern!  Another question is: we wrote our group operations as permutations of the letters A,B,C,D above.  But there were only 8 group operations, and there are 24 permutations total.  Which permutations couldn’t we do, and why not?

1…and who also led a heroically tragic life, getting rejected from one academy because his ideas were too dense for the instructors to understand, expelled from another for his radical politics, and writing his most important work the night before he died in a duel at the age of 21.

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