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Permutation and Dihedral Groups

Michael Freeze

MAT 541: Modern Algebra IUNC Wilmington

Fall 2013

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Overview

Starting Definitions

Factorizations of Permutations

Cycle Structure of Permutations

Orders of Permutations

Parity of Permutations

Dihedral Groups

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Outline

Starting Definitions

Factorizations of Permutations

Cycle Structure of Permutations

Orders of Permutations

Parity of Permutations

Dihedral Groups

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Definition of Permutation

DefinitionA permutation of a set X is a bijection from X to itself.

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Definition of Symmetric Group

DefinitionThe family of all the permutations of a set X , denoted by SX ,is called the symmetric group on X .

When X = {1, 2, . . . , n}, SX is usually denoted by Sn, and it iscalled the symmetric group on n letters.

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Definition of Cycle

DefinitionLet i1, i2, . . . , ir be distinct integers in {1, 2, . . . , n}. If α ∈ Sn

fixes the other integers (if any) and if

α(i1) = i2, α(i2) = i3, . . . , α(ir−1) = ir , α(ir ) = i1,

then α is called an r -cycle. We also say that α is a cycle oflength r , and we denote it by

α = (i1 i2 . . . ir ).

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Disjoint Permutations

DefinitionTwo permutations α, β ∈ Sn are disjoint if every i moved byone is fixed by the other: If α(i) 6= i , then β(i) = i , and ifβ(j) 6= j , then α(j) = j . A family β1, . . . , βt of permutations isdisjoint if each pair of them is disjoint.

LemmaIf β moves i , then β moves β(i).

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Disjoint Permutations Commute

PropositionDisjoint permutations in Sn commute.

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Outline

Starting Definitions

Factorizations of Permutations

Cycle Structure of Permutations

Orders of Permutations

Parity of Permutations

Dihedral Groups

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Complete Factorization

PropositionEvery permutation α ∈ Sn is either a cycle or a product ofdisjoint cycles.

DefinitionA complete factorization of a permutation α is a factorizationof α into disjoint cycles that contains exactly one 1-cycle (i)for every i fixed by α.

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Inverses of Cycles and Products

Proposition

(i) The inverse of the cycle α = (i1 i2 . . . ir ) is the cycle(ir ir−1 . . . i1) :

(i1 i2 . . . ir )−1 = (ir ir−1 . . . i1).

(ii) If γ ∈ Sn and γ = β1 · · · βk , then

γ−1 = β−1k · · · β

−11 .

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Complete Factorization is Essentially Unique

TheoremLet α ∈ Sn and let α = β1 · · · βt be a complete factorizationinto disjoint cycles. This factorization is unique except for theorder in which the cycles occur.

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Outline

Starting Definitions

Factorizations of Permutations

Cycle Structure of Permutations

Orders of Permutations

Parity of Permutations

Dihedral Groups

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Cycle Structure

DefinitionTwo permutations α, β ∈ Sn have the same cycle structure iftheir complete factorizations have the same number of r -cyclesfor each r .

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Conjugates have the same Cycle Structure

LemmaIf γ, α ∈ Sn, then αγα−1 has the same cycle structure as γ. Inmore detail, if the complete factorization of γ is

γ = β1β2 · · · (i1 i2 . . .) · · · βt ,

then αγα−1 is the permutation that is obtained from γ byapplying α to the symbols in the cycles of γ.

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Permutations with same Cycle Structure are

Conjugates

TheoremPermutations γ and σ in Sn have the same cycle structure ifand only if there exists α ∈ Sn with σ = αγα−1.

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Outline

Starting Definitions

Factorizations of Permutations

Cycle Structure of Permutations

Orders of Permutations

Parity of Permutations

Dihedral Groups

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Orders of Permutations

PropositionLet α ∈ Sn.

(i) If α is an r -cycle, then α has order r .

(ii) If α = β1 · · · βt is a product of disjoint ri -cycles βi , thenα has order lcm(r1, . . . , rt).

(iii) If p is a prime, then α has order p if and only if it is ap-cycle or a product of disjoint p-cycles.

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Outline

Starting Definitions

Factorizations of Permutations

Cycle Structure of Permutations

Orders of Permutations

Parity of Permutations

Dihedral Groups

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Permutations are Products of Transpositions

PropositionIf n ≥ 2, then every element of Sn is a product oftranspositions.

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Parity of Permutations

DefinitionA permutation α ∈ Sn is even if it can be factored into aproduct of an even number of transpositions; otherwise, α isodd. The parity of a permutation is its status as even or odd.

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Definition of Signum Function

DefinitionIf α ∈ Sn and α = β1 · · · βt is a complete factorization intodisjoint cycles, then signum α is defined by

sgn(α) = (−1)n−t .

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Signum is Multiplicative

TheoremFor all α, β ∈ Sn,

sgn(αβ) = sgn(α)sgn(β).

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Signum and Parity

Theorem

(i) Let α ∈ Sn; if sgn(α) = 1, then α is even, and ifsgn(α) = −1, then α is odd.

(ii) A permutation α is odd if and only if it is a product of anodd number of transpositions.

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Example

Alternating GroupThe subset An = {α ∈ Sn : α is even } of Sn is a subgroup,called the alternating group on n symbols.

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Outline

Starting Definitions

Factorizations of Permutations

Cycle Structure of Permutations

Orders of Permutations

Parity of Permutations

Dihedral Groups

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Dihedral Groups

DefinitionThe group of symmetries of an n-sided regular polygon iscalled the dihedral group of order 2n, and is denoted by D2n.

The group D2n contains 2n elements, namely the rotationsI,R ,R2, . . . ,Rn−1 and the reflections F ,FR ,FR2, . . . ,FRn−1.

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D6 versus S3

Note that both D6 and S3 have order 6, with half of theelements of order 2.

In what other ways are the two groups related?

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Review

Starting Definitions

Factorizations of Permutations

Cycle Structure of Permutations

Orders of Permutations

Parity of Permutations

Dihedral Groups

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