Let G be a group of permutations of a set S. Prove that the orbits of the members of S constitut

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Let G be a group of permutations of a set S. Prove that the...

Key Concepts

Partition

A partition of a set is a collection of non-empty, mutually disjoint subsets whose union is the entire set. In the scenario of a group acting on a set, the orbits serve as these subsets because they are both disjoint and collectively exhaustive. Demonstrating that the orbits of a group action form a partition is crucial as it confirms that every element of the set is accounted for in exactly one orbit, thereby structuring the set in a well-organized manner.

Equivalence Relation

An equivalence relation on a set is a binary relation that is reflexive, symmetric, and transitive. In the context of group actions, the relation that groups two elements if one can be transformed into the other by some group element satisfies these properties. Consequently, this equivalence relation partitions the set into distinct classes where each class is an orbit, ensuring that each element of the set belongs to exactly one orbit.

Group Action

A group action is a formal way in which a group, consisting of elements that satisfy certain algebraic properties, interacts with a set. When a group acts on a set, each group element 'moves' or 'permutes' the elements of the set in a systematic way that respects the group structure, meaning that the identity element maps every element to itself and the action of a product of two group elements is equivalent to successive actions of each element. This notion is fundamental in studying symmetries and transformations.

Orbit

The orbit of an element in a set under a group action is the collection of all elements that can be reached by applying every group element to that particular element. In other words, it is the set of images of the element under the action of all elements of the group. Orbits capture the idea of how group actions spread a single element across the set and are central in understanding how the set is decomposed under the symmetry provided by the group.

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