Quiz in Permutation Group and Cosets

What is the symmetric group S₆​?

• A group of all 6x6 matrices

• The group of all permutations of 6 elements

• A group with only the identity element

• The set of all even integers

Which of the following is the identity permutation in array form for S₅?

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A permutation σ is its own inverse if:

• σ = identity

• σ is even

• σ is odd

• σ² = identity

What is the result of the binary operation "composition" on permutations α = (1 2) and β = (2 3) in S₃?

• (1 2) (2 3)

• (1 3)

• (1 2 3)

• (1 3 2)

The orbit of an element under a group action is

• The identity element

• The set of elements fixed by the permutation

• The set of elements that an element can be mapped to

• Always a singleton

Which of the following statements about even and odd permutations is true?

• Every permutation is even.

• Even permutations are those that can be expressed as an odd number of transpositions.

• The identity permutation is even.

• An even permutation has no inverse.

How many left cosets are there of a subgroup of order 3 in S₃?

• 6

• 3

• 2

• 1

Which of the following permutations in S₄​ is a product of disjoint cycles?

• (1 2) (2 3)

• (1 3) (2 4)

• (1 2 3)

• (1 4 2)

Let σ = (1 2 3) (4 5). What is σ ^-1 in cyclic form?

• (1 2 3) (4 5)

• (3 2 1) (4 5)

• (1 3 2) (5 4)

• (3 1 2) (5 4)

Find the product of the transpositions (2 3) (1 3) (5 6) (6 7) (4 7).

• (1 2 3) (5 4 6 7)

• (1 3 2) (4 5 6 7)

• (1 2 3) (4 5 6 7)

• (1 2 4) (3 6 5 7)