Groups, Transformations, and Representations
1. Writing down a Group
- Multiplication table
a multiplication table records the product of any two group elements. A typical multiplication table looks like the following:
\ e g1 g2 g3 … e e g1 g2 g3 … g1 g1 g12 g1 g2 g1 g3 … g2 g2 g2 g1 g22 g2 g3 … g3 g3 g3 g1 g3 g2 g32 … \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\ddots\) Since we dont't know whether the group is Abelian or not, gi gj ≠ gj gi in genernal.
- The rearrangement lemma
Given any sequece of all the group elements of a group, multiplying the sequence by the same group element simply reorders the sequence. I.e., \(g\cdot G = G\), \(G\cdot g = G\).
E.g.: Now the groups \(\mathbb{Z}_4: (\{0, 1, 3, 4\}, (a+b)\mod 4)\) and \(G_i\) are the same group in the sense that they are defined by the same 4 × 4 multiplication table below:
\ e A B C e e A B C A A B C e B B C e A C C e A B - Presentation of a group
- A arbitrary discrete group can always be uniquely defined by a set of generators and a set of relations on the generators. Typically, we can write \[G = (x_1, x_2, \dots, x_m; r_1, r_2, \dots, r_n),\] which is called the presentation of \(G\). E.g.: \(\mathbb{Z}_4\cong G_i = (a; a^4 = 1).\)
2. A Simple Transformation Group and Its Represnetation
A linear representation of group G is a pair (ρ, V), where V is a vector space (finite of infinite dims) over \(\mathbb{R}\) or \(\mathbb{C}\), and \( \rho: G\mapsto \text{End}(V) \). We called V the representation space.
If ρ is injective, (ρ, V) is faithful represnetation.
Two linear represnetation (ρ, V) and (ρ', V) of G are equivalent, if and only if there exists a similarity transformation, i.e., a constant matrix S ∈ GL(V) such that S ρ(g) S-1 = ρ'(g), ∀ g ∈ G.
3. Subgroups and quotient groups
A subgroup H of group G is a subset of G and is closed under the group multiplication of G. If H = {e} or H = G, it's a trival subgroup; otherwise, it's a proper subgroup.
For finite groups the following theorem relates H and G if H is a subgroup of G.
- Lagrange's theorem: for a finite group, |H| divdes |G|, i.e, |G| = 0 (mode |H|). In other words, \[\frac{|G|}{|H|} = n \in \mathbb{N}^+.\]
If g ∈ G \ H, in general gH ≠ Hg. When H satisifies gH = Hg, ∀ g ∈ G, we say H is an invariant/normal subgroup of G \((H\triangleleft G)\). Obviously, H = g-1Hg = gHg-1.
If a group G has no invariant subgroup, is said to be simple simple group.
If (H\triangleleft G), then G/H = {gH | g ∈ G} is called quotient group of G by H.