(5) Permutations to Transpositions

Question 1

Define permutation of a nonempty set A

Answer 1

a function from A to A that is a bijection

Question 2

T/F. Function composition is a binary operation on the set of permutation of A.

Answer 2

T

Question 3

The direction of the composition of permutations

Answer 3

Right to left

Question 4

σ∘τ≠τ∘σ. Why?

Answer 4

composition of permutation is not commutative unless they are disjoint

Question 5

If |A|=|B|, then

Answer 5

SsubA ≅ SsubB

Question 6

_____ is the group of permutations of A under composition

Answer 6

Symmetric Group of Degree n, Ssubn

Question 7

What is D3?

Answer 7

group of symmetries of an equilateral triangle

Question 8

T/F. If n≥3, then Sn is abelian,

Answer 8

F. non cyclic => not non abelian

Question 9

State Cayley’s Theorem

Answer 9

Every group is isomorphic to a group of permutation

Question 10

What is the left regular representation of G?

Answer 10

xxxxxxxxxxxxxxx

Question 11

What is the right regular representation of G?

Answer 11

xxxxxxxxxxxxxxxxxxx

Question 12

The relation ∽ defined by x∽y is an equivalence relation on A iff

Answer 12

y=σ^k(x) for some k ∈ Z

Question 13

Let σ ∈ Ssubn and x ∈ A={1,2,3,…,n}. The orbit σ containing x is

Answer 13

{σ^k(x) | k∈ℤ}

Question 14

The orbits of σ ∈ Ssubn forms a partition on Set A. Why»

Answer 14

Because ∽ is an equivalence relation

Question 15

What is a cycle?

Answer 15

A permutation σ ∈ Ssubn if it has at most 1 orbit containing more than one element.

Question 16

What is the length of a cycle?

Answer 16

number of elements in its largest orbit?

Question 17

idsubA=?

Answer 17

(1)

Question 18

T/F. The cycle notation of any permutation is not unique.

Answer 18

T

Question 19

What is the inverse of a cycle: (a1,a2,a3,…,an)?

Answer 19

(a, an, an-1, an-2,…,a2)

Question 20

inverse of α∘β?

Answer 20

β^-1∘α^-1

Question 21

The order of a cycle is its ___

Answer 21

length

Question 22

Let σ1,σ2,,..,σk be cycles of Ssubn. We say that the cycles are DISJOINT if for all ________ and ________, σi(a)_____ implies that _______.

Answer 22

i ∈ {1,2,3,..,k} and a ∈ A ={1,2,3…,n}

σi(a≠a implies that σj(a)=a for each j ∈ {1,2,3,..,k}{i}

Question 23

T/F. Any permutation can be written as a cycle but not as a product of disjoint cycles.

Answer 23

F. Both true.

Question 24

If σ,τ are disjoint cycles then

Answer 24

σ∘τ=τ∘σ.

Question 25

(12)(34)(56)=(34)(12)(56). Why?

Answer 25

unique up to the arrangement of the factors

Question 26

The order of σ is the ____ of the disjoint cycles whose product is σ

Answer 26

LCM of the lengths

Question 27

A cycle of length 2

Answer 27

transposition

Question 28

T/F. Every permutation σ ∈ Ssubn is a product of transpositions.

Answer 28

F. n≥1

Question 29

The inverse of a transposition is

Answer 29

itself

Question 30

The identity permutation is an even permutation. Why>

Answer 30

Because the total number of transpositions of any σ ∈ Ssubn and its inverse is always even.

Question 31

T/F. A permutation can be written as a product of even number and at the same time odd number of permutations

Answer 31

F

Question 32

If σ ∈ Ssubn is a cycle of length k, then σ is an even permutation iff

Answer 32

k is odd that is the number of transpositions is k-1

Question 33

If α,β are even permutations then α∘β must be ____

Answer 33

also even

Question 34

T/F. The set of even permutations in Ssubn forms a subgroup if Ssubn.

Answer 34

T

Question 35

subgroup of Ssubn containing all even permutations of Sn

Answer 35

alternating group of degree n, Asubn

Question 36

The number of even and odd permutations are equal. That is,

Answer 36

|Asubn|=n!/2