Cycle notation in the context of Automorphism

Cycle notation in the context of Automorphism

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⭐ Core Definition: Cycle notation

In mathematics, a permutation of a set can mean one of two different things:

an arrangement of its members in a sequence or linear order, or
the act or process of changing the linear order of an ordered set.

An example of the first meaning is the six permutations (orderings) of the set {1, 2, 3}: written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory.

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👉 Cycle notation in the context of Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

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⭐ Core Definition: Cycle notation
👉 Cycle notation in the context of Automorphism