Section 9 Orbits, Cycles, and The Alternating Groups

9.1 Definition Orbit

p.87

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

p.89

A permutation $\sigma \in S_n$ 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 $\sigma$: it shows how $\sigma$ sends each element “in a loop” among a small subset of $\{1,2,\dots ,n\}$.
• It is part of the notation or decomposition of $\sigma$ inside $S_n$​.

9.8 Theorem

p.89

Every permutation σ of a finite set is a product of disjoint cycles.

9.11 Definition

p.90

A cycle of length 2 is a transposition.

9.15 Theorem

p.91

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

p.93

The subgroup of $S_n$ consisting of the even permutations of n letters is the alternating group $A_N$ on n letters.