Group Theory¶

Basic Concepts¶

A Group is a structure consisting of a set extended with an operation which together satisfy four axioms.

Given two groups, \((G,*)\) and \((H, \cdot)\), a group isomorphism from \((G,*)\) to \((H,\cdot)\) is a bijective group homomorphism from \(G\) to \(H\), that is, an isomorphism is a bijective function \(f : G \to H\), such that, for all \(u,v \in G\), it holds

\[f(u*v) = f(u) \cdot f(v)\]

If such a function exists we can write

\[(G,*) \cong (H,\cdot)\]

The group of real numbers under addition, and the group of real numbers under multiplication are isomorphic under the bijection

[Abelian Group] A group, \((G,*)\) is called Abelian, if, in addition to the axioms for a group, it also satisfies a commutative property;

5. For all \(a,b \in G\), \(a*b = b*a\).

[Subgroup] A set \(H\) which is a subset of \(G\), where \((G,*)\) is a group, is called a subgroup iff \((H,*)\) is a group, and

A subgroup, \(H\), is a trivial subgroup if the group, \(G\), has only the identity element. Otherwise, if \(H \neq G\) then \(H\) is a proper subgroup.

[Generating Set] The generating set of a group is a subset such that, any element of the group can be expressed as the combination of finitely many elements of the subset and their inverses.

[Cayley Diagram] Suppose that \(G\) is a group, and \(S\) is a generating set for \(G\). They Cayley diagram, \(\Gamma = \Gamma(G,S)\) is a coloured directed graph with the construction:

• each element \(g\in G\) is assigned a vertex. The vertex set \(V(\Gamma)\) of \(\Gamma\) is thus identified with \(G\).

• each generator is assigned a colour, \(c_s\)

• For any \(g\in G, s \in S\), the vertices corresponding to the elements \(g\) and \(gs\) are joined by a directed edge of colour \(c_s\), and thus the edge set, \(E(\Gamma)\) is composed of the pairs of form \((g,gs)\), with \(s\in S\) providing the colour.

For simplicity, the identity element is omitted, leaving a normal graph without loops.

Finite Groups¶

Continuous Groups¶

The continuous, or Lie groups, are groups which are composed of an infinite set equipped with a binary operation. Lie groups are also differentiable manifolds, with the property that the group operation is compatible with the smooth structure of the manifold. They are named after Sophus Lie, who laid the foundations for their study. Of particular interest to physics are the classical groups, all of which are closely related to symmetry in Euclidean spaces. There are seven classical groups;

• general linear—GL(\(n\))

• special linear—SL(\(n\))

• orthogonal—O(\(n\))

• special orthogonal—SO(\(n\))

• unitary—U(\(n\))

• special unitary—SU(\(n\))

• symplectic—Sp(\(n\))