Groups
A key insight into the nature of groups involves recognizing their relation to monoids: Groups are monoids with the property that every element has an inverse.
Monoids are likely useful in thinking about diffusion, in which time plays a role and things cannot be undone. Groups are more likely useful in thinking about mechanics, where actions are time-reversible.
Given this definition alone, one might immediately think of playing with a Rubik's cube as an example of groups at work. Twist, twist, twist. Whoops. Untwist, untwist, untwist.
Fact: An element m in M of a Monoid (M, e, ⋆) has at most one inverse. This follows from the associative law for monoids.
The monoid (ℕ, 0, +) is not a group, but (ℤ, 0, +) is a group. Every element of the monoid built from the integers has an inverse (the one built from the natural numbers sees only the unit element as having an inverse). This gives rise to the definition of a group:
A group is a monoid (M, e, ⋆) in which every element m ∈ M has an inverse.
Groups are all about symmetries.
{ e,ρ,ρ2,ρ3,ϕ,ϕρ,ϕρ2,ϕρ3 }
where
ρ = clockwise 90 degrees
ϕ flip across vertical axis
All isometries of ℝ3, i.e., all functions ℝ3 → ℝ3 that preserve distances.
Iso(X) ≔ {f : X → X | f is an isomorphism} is a group.
For any point x ∈ X, the orbit of x, denoted Gx, is the set
Gx≔{ x′∈X|∃g∈G such that gx=x′ }.
If G is the rotational group acting on the surface S of the globe (where we rotate about the vertical axis), then for a point x on the surface S, its orbit is the latitude line running through x.
If y is in the orbit of x, then x is in the orbit of y (symmetry). Also, x is in the orbit of itself, since a group has the unit element e (reflexivity). If x is in the orbit of y and z is in the orbit of y, then z is in the orbit of x (transitivity). Being in the same orbit (on X) is an equivalence relation on X.