By Omer Cabrera

Desk of Contents

Chapter 1 - Symmetry

Chapter 2 - team (Mathematics)

Chapter three - team Action

Chapter four - standard Polytope

Chapter five - Lie aspect Symmetry

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**Sample text**

There is no x such that x · 0 = 1), (Q, ·) is still not a group. However, the set of all nonzero rational numbers Q \ {0} = {q ∈ Q, q ≠ 0} does form an abelian group under multiplication, denoted (Q \ {0}, ·). Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is b/a, therefore the axiom of the inverse element is satisfied.

This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The quotient X/G inherits the quotient topology from X, and is called the quotient space of the action. The above statements about isomorphisms for regular, free and transitive actions are no longer valid for continuous group actions.

Definition If G is a group and X is a set, then a (left) group action of G on X is a binary function denoted which satisfies the following two axioms: 1. (gh)·x = g·(h·x) for all g, h in G and x in X; 2. e·x = x for every x in X (where e denotes the identity element of G). The set X is called a (left) G-set. The group G is said to act on X (on the left). From these two axioms, it follows that for every g in G, the function which maps x in X to g·x is a bijective map from X to X (its inverse being the function which maps x to g-1·x).