Virtual Labs

Permutations and Permutation Groups

For a set

X=\{1,2,3,4,5,6,7,8\}

and a permutation with image sequence

[4,2,1,3,6,5,8,7]

(where position

i

maps to the

i

-th value), which is the longest cycle in this permutation?

(1,4,3)

Explanation

(5,6)

Explanation

(7,8)

Explanation

(2)

Explanation

For a set

X=\{1,2,3,4,5,6,7,8\}

and its permutation

Y=(4,2,1,3,6,5,8,7)

, the number of cycles including 1-cycles in

Y

are

2

Explanation

4

Explanation

5

Explanation

d: none of the above Explanation

Explanation

If there are

a

number of solvable instances of the 15-puzzle, how many of the 15-puzzle instances are unsolvable?

\frac{a}{2}

Explanation

a

Explanation

2a

Explanation

d: none of the above Explanation

Explanation

For a set

X={1,2,3,4,5,6,7,8}

and its permutation

Y=(4,2,1,3,6,5,8,7)

, the given permutation is odd.

a: true Explanation

Explanation

b: false Explanation

Explanation

Let there be some permutation

x

of the set

X

. We are given that

x^2=a

x^3=b

x^4=x

and

x^5=a

where

a,b,x

are permutations of

X

. What is the order of

x

3

Explanation

4

Explanation

5

Explanation

d: none of the above Explanation

Explanation

What is the cycle decomposition of the permutation

(1,4,2)(3,5,6,7)

(1,2)(2,4)(3,5)(5,6)(6,7)

Explanation

(1,4)(4,2)(3,5)(5,6)(6,7)

Explanation

(1,2)(2,4)(3,7)(7,6)(6,5)

Explanation

(1,4)(4,2)(3,7)(7,6)(6,5)

Explanation

What is the minimum number of 2-cycles needed to represent the permutation

(1,2,3,4,5)(6,7)

a: 3 Explanation

Explanation

b: 4 Explanation

Explanation

c: 5 Explanation

Explanation

d: 6 Explanation

Explanation

If a permutation

p

has cycle structure

(1,2,3)(4,5,6,7)

, what is

(p^2)^3

a: Identity permutation Explanation

Explanation

p^6

Explanation

p^3

Explanation

p^4

Explanation

Given permutations

α=(1,2,3,4)

and

β=(2,3,4,5)

in cycle notation, what is the order of

α∘β

a: 4 Explanation

Explanation

b: 5 Explanation

Explanation

c: 20 Explanation

Explanation

d: 6 Explanation

Explanation

In a 15-puzzle, if the current state has odd parity and the empty space is in an even position (counting from top-left, row by row), what can you conclude?

a: The puzzle is always solvable Explanation

Explanation

b: The puzzle is never solvable Explanation

Explanation

c: Solvability depends on the number of inversions Explanation

Explanation

d: Need more information to determine solvability Explanation

Explanation

What is the sign of the composition

(1,2,3)(4,5) ∘ (2,3,4)(1,5)

a: Even Explanation

Explanation

b: Odd Explanation

Explanation

c: Depends on the order of composition Explanation

Explanation

d: Cannot be determined without expanding Explanation

Explanation