I need to come up with a geometric example of a group. I've been given an example, the set of all symmetries of an equilateral triangle. But I then have to show that the axioms are satisfied and I dont really understand how they are.
I know the axioms are the following, given a set X and an operation *: (X x X) --> X then (X,*) is a group if:
(1) there exists a neutral element e such that x*e=e*x=x for all x in X
(2) the operation is associative
(3) every element x in X has an inverse y such that x*y=y*x=e
The set of all symmtries of an equilateral triangle contains
six elements : three line symmtries, and three rotational symmtries;
It is actually the collection of all the transformation which keep the equilateral triangle remain unchange.
It is easy to verify that the three axioms of Group is satisfied.
The operation is the composition of transform(mapping).
the group where I is the identity transform, R is the Rotational Transformation of 120 degree with respect to the center of the equilateral triangle, T is the symmetrical transformation with respect to one(fixed) of the three symmetry line, and .
So,for example,the inverse of R is [tex]R^2[tex], the inverse of T is T itself.
the composition of transformation obviously satisfies the associate law.
It is very easy to virefy that all axioms of group are satisfied, you can do this by write down the operation table of G, then you can understand the group better !