To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science, Albert Einstein.
Proposition. Suppose G is a group. If ∀a, b ∈ G (ab)^{2} = a^{2}b^{2}, then ab = ba.
Proof.
a(ab)b = a^{2}b^{2} = [By assumption] (ab)^{2} = (ab)(ab) = [Associative] a(ba)b
a(ab)b = a(ba)b ⇒ [Cancellation laws ↭ multiplying both sides by a^{-1}, and then, by b^{-1}] ab = ba∎
Proposition. If every non-identity element of G has order 2, then G is Abelian.
Proof.
G is Abelian ↭ ∀a, b ∈ G, ab = ba.
∀a, b ∈ G ⇒ [G group, it is closed under the group operation] ab ∈ G ⇒ [∀a ∈ G, a^{2} = e, notice that e^{2}= e, too] (ab)^{2} = e ⇒ ab = (ab)^{-1} = [Shoes and socks principle] b^{-1}a^{-1} = [Every element is its own inverse] ba∎
A Cayley table describes the structure of a finite group by arranging all the possible products of all the group’s elements in a square table.
Let G = {a_{1}, a_{2}, ···, a_{n}}. The group operation (e.g., +, x, *, ·, ∘) is placed in the upper left corner of the table. Next, we place the elements of the set in the head row and the head column. The i, j entry, that is, the intersection of row i and column j corresponds to the group element a_{i} · a_{j}. For example, let’s say the group G ({1, -1, i, -i}, x), -1·-1 = 1, -1·i = -i, -1·-i = i, etc.
Let’s take into consideration the Cayley table for G ({1, -1, i, -i}, x).
Indeed group theory is the mathematical language of symmetry. We can think groups as symmetries (a transformation that preserves both distances and the object itself). Futhermore, symmetries of an object (a plane figure -square-, a solid figure -tetrahedron- or a real number line ℝ) do form a group (💡Source: Mathemaniac YouTube’s channel, Essence of Group Theory).
Definition. A group action of a group G on a set A, written as g·a or simply ga (g acts on a), is a map or function from G x A to A such that the following two conditions hold:
Proposition. Let the group G act on a set A. For each fixed element of G, say g ∈ G, we get a permutation σ_{g}: A → A defined by σ_{g}(a) = g·a. Besides, the map from Φ: G → S_{A} defined by g → σ_{g} is a homomorphism. It is called the permutation representation associated to the given action.
Proof. We know that the map f: A → B is a bijection if and only if there exists g: B → A, the inverse of f, such that f∘g is the identity map on B and g∘f is the identity map on A.
∀g ∈ G, ∃g^{-1}, we claim that σ_{g-1} is the inverse of σ.
(σ_{g-1}∘σ_{g})(a) = [By definition of function composition] σ_{g-1}(σ_{g}(a)) = σ_{g-1}(g·a) = g^{-1}·(g·a) = [By property 1 of a group action] (g^{-1}g)·a = e·a = [By property 2 of a group action] a
Since g was chosen arbitrary, we may interchange the roles of g and g^{-1} ⇒ σ_{g} is a permutation of A.
is the map Φ: G → S_{A} a homomorphism? We have already proved that σ_{g} ∈ S_{A}.
Φ(g_{1}g_{2}) = Φ(g_{1})∘Φ(g_{2})?
∀a ∈ A, Φ(g_{1}g_{2})(a) = σ_{g1g2}(a) = (g_{1}g_{2})·a = [By property 1 of an action] g_{1}·(g_{2}·a) = σ_{g1}(σ_{g2}(a)) = (Φ(g_{1})∘Φ(g_{2}))(a)
Basically, a group action just means that every element g in G acts as a permutation on A in a way that is consistent with the group operation in G. Let Φ: G → S_{A} be any homomorphism from a group G to the symmetric group on a set, then the map from G x A to A defined by ∀g ∈ G, a ∈ A, g·a = Φ(g)(a) is a group action of G on A, and therefore actions on a group and homomorphisms from this group into the symmetric group S_{A} are basically the same construct or, more formally, in bijective correspondence.
Examples:
Definition. If G acts on a set A and distinct elements of G induce distinct permutations of A, that is, when g_{1} ≠ g_{2} in G, there is an a ∈ A such that g_{1}·a ≠ g_{2}·a, the action is called faithful. The kernel of the action of G on A is {g ∈ G | ga = a ∀a ∈ A}
The symmetric group (G =) S_{A} acting on A, S_{A} x A → A, by σ·a = σ(a) ∀σ ∈ S_{A}, a ∈ A. The associated permutation representation Φ: S_{A} → S_{A} is the identity map.
We can make ℝ^{n} act on itself by translations, for v ∈ ℝ^{n}, let T_{v}: ℝ^{n} → ℝ^{n}, defined by T_{v}(w) = w + v. The axioms for a group action are: T_{0}(w) = w and T_{v1}(T_{v2}(w)) = T_{v1+v2}(w). These are true properties of vector addition: w + 0 = w, (w + v_{2}) + v_{1} = w + (v_{1} + v_{2}).
For n ≥ 3, D_{n} acts on a regular n-gon as rigid motions. We can also view D_{n} as acting just on the n vertices of a regular n-gon. This does not lose any information at all, since knowing where each and every vertex go under a rigid motion determines completely where everything else goes. By focusing on the action of D_{n} on the n vertices, and labelling them by 1, 2, ···, n, we make D_{n} act on {1, 2, ···, n} (See the previous illustration).
We can make G act on itself by left multiplication. We let A = G and g·a = ga is merely the usual product of g and a in G.
Let G = ℤ/4ℤ act on itself by additions. For instance, addition by 1 has the effect 0 → 1 → 2 → 3 → 0. Addition by 1 on ℤ/4ℤ is a 4-cycle (0123). Addition by 2 has the effect 0 → 2 → 4, 1 → 3, it is a product of two 2-cycles, (02)(13). The composition of these two permutations is (0123)(02)(13) = (0321) which is the permutation of G described by addition by 3.