Let's Study Real Geometry
Groups
Let's begin with the definition of a group. We will write our groups as (G, *) where G is a set (this is just a collection of things) and * is a binary operation on G. This means that * takes two elements of G and returns a third. Furthermore, (G, *) must satisfy three axioms:
- associativity: for all a, b and c in G, (a*b)*c = a*(b*c)
- identity element: there exists an element e in G such that for all elements a of G, e*a = a
- existence of inverses: For every element a of G, there exists an element which we write as a^-1 or as -a such that a*a^-1 = e.
Examples of Groups
- (Z, +)
Z is the set of integers. + is the usual addition of integers. 0 is the identity element, if a is an integer than -a is it's inverse as a + (-a) = 0.
- (R+, *)
R+ is the set of real numbers x such that x > 0. * is the usual multiplication of real numbers. 1 is the identity and x^-1 = 1/x. Then x*1/x = 1
- (Z[X], +)
Z[X] is the set of polynomials in the variable X with integer coefficients. + is the component wise addition of polynomials. For example, if f = anXn + ... + a1X + a0 and g = bnXn + ... + b1X + b0, then f + g = (an + bn)Xn + ... + (a1 + b1)X + (a0 + b0). If terms are missing, then just fill in with zeros. One can add polynomials of any degree. Then the identity element is just the zero polynomial with all of it's coefficients equal to zero. If f has coefficients ai then the coefficients of -f are -ai. Then f + (-f) = 0 where zero in this context is the zero polynomial and not the number zero.
The Dihedral Groups
Now we can get to a little geometry. Let's define a symmetry of a shape to be something that I can do to the shape without changing it. It's not too formal but I can clean it up a little if anybody wants. For example, if I have a circle and I rotate it around it's center, that is a symmetry of the circle: unless some mark is placed on the circle, there is no way to know that it has been rotated. Given a geometric shape and the set of it's symmetries, lets define a binary operation. If a and b are symmetries of our shape (let's call it C), then if I do a and then do b that is a symmetry well. This operation is called composition. For example, if I have a circle and I rotate it 30 degrees and then rotate it -10 degrees, it's like I rotated it 20 degrees.
This operation and the set of symmetries form a group. I'll leave the verification of the details as an exercise: real math is not a spectator sport.
The n-th Dihedral group is the set of symmetries of the regular n-gon along with the binary operation of composition. Let's call the group (Dn, *). Can you find all of it's elements if we take n to be 3?
If there is any curiosity, we can continue.