9.1 Definition Orbit
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Let σ be a permutation of a set A. The equivalence classes in A determined by the equivalence relation (1) are the orbits of σ .
1→3→6→1→…
- Depends on a whole group G: OrbG(x) collects every image of x under every element of the group.
- It is a subset of the underlying set X on which G acts.
9.6 Definition Cycle
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A permutation is a cycle if it has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit.
- Describes a single permutation : it shows how sends each element “in a loop” among a small subset of .
- It is part of the notation or decomposition of inside .
9.8 Theorem
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Every permutation σ of a finite set is a product of disjoint cycles.
9.11 Definition
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A cycle of length 2 is a transposition.
9.15 Theorem
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No permutation in Sn can be expressed both as a product of an even number of transpositions and as a product of an odd number of transpositions.
Think about a determinant. Exchange any two rows of a square matrix changes the sign of the determinant.
A Permutation that can be expressed as a product of an even number of transpositions is called a even permutation. Similar to a odd permutation.
9.21 Definition Alternating Group
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The subgroup of consisting of the even permutations of n letters is the alternating group on n letters.