Permutations

Question 1

What is a permutation in the context of group theory?

Answer 1

A permutation is a rearrangement of the elements of a set.

Question 2

True or False: A group is a set equipped with an operation that satisfies closure, associativity, identity, and invertibility.

Answer 2

True

Question 3

What is the symmetric group on n elements denoted as?

Answer 3

It is denoted as S_n.

Question 4

Fill in the blank: The number of permutations of n distinct objects is _____ .

Answer 4

n!

Question 5

Define a cyclic permutation.

Answer 5

A cyclic permutation is a permutation where a group of elements is rotated among themselves.

Question 6

What is the order of a permutation?

Answer 6

The order of a permutation is the smallest positive integer k such that applying the permutation k times returns to the original arrangement.

Question 7

True or False: Every permutation can be expressed as a product of transpositions.

Answer 7

True

Question 8

What is a transposition?

Answer 8

A transposition is a permutation that swaps two elements and leaves all others unchanged.

Question 9

What does it mean for a permutation to be even?

Answer 9

A permutation is even if it can be expressed as a product of an even number of transpositions.

Question 10

What does it mean for a permutation to be odd?

Answer 10

A permutation is odd if it can be expressed as a product of an odd number of transpositions.

Question 11

What is the notation for the alternating group of degree n?

Answer 11

It is denoted as A_n.

Question 12

How many elements are in the symmetric group S_n?

Answer 12

S_n has n! elements.

Question 13

What is the relationship between S_n and A_n?

Answer 13

A_n is a subgroup of S_n consisting of all even permutations.

Question 14

Define the concept of the cycle structure of a permutation.

Answer 14

The cycle structure is the decomposition of a permutation into disjoint cycles.

Question 15

What is a k-cycle?

Answer 15

A k-cycle is a cycle that permutes k elements while fixing all other elements.

Question 16

What is the order of a permutation given its cycle structure?

Answer 16

The order is the least common multiple of the lengths of the cycles in the cycle structure.

Question 17

True or False: The identity permutation is an element of every symmetric group.

Answer 17

True

Question 18

What is the effect of a permutation on the elements of a set?

Answer 18

The permutation rearranges the elements of the set.

Question 19

Fill in the blank: The inverse of a permutation is a permutation that _____ the original permutation.

Answer 19

reverses

Question 20

What is a permutation group?

Answer 20

A permutation group is a group whose elements are permutations of a given set.

Question 21

What is the symmetric group S_3?

Answer 21

S_3 is the group of all permutations of three elements.

Question 22

How many elements does A_3 have?

Answer 22

A_3 has 3 elements.

Question 23

What is the Cayley’s theorem in the context of permutation groups?

Answer 23

Cayley’s theorem states that every group is isomorphic to a subgroup of a symmetric group.

Question 24

What is the stabilizer of an element in a group action?

Answer 24

The stabilizer is the set of permutations that leave that element unchanged.

Question 25

Define a group action.

Answer 25

A group action is a way of describing how a group interacts with a set.

Question 26

What is a coset in group theory?

Answer 26

A coset is a form of partitioning a group formed by multiplying all elements of a subgroup by a fixed group element.

Question 27

What is the orbit of an element under a group action?

Answer 27

The orbit is the set of elements that can be reached by applying the group elements to that element.

Question 28

Fill in the blank: The kernel of a group action is the set of all elements that _____ every element of the set.

Answer 28

fix

Question 29

True or False: The symmetric group S_n is non-abelian for n ≥ 3.

Answer 29

True

Question 30

What is a complete group in terms of permutations?

Answer 30

A complete group is a group where every automorphism is inner.

Question 31

What is the significance of the sign of a permutation?

Answer 31

The sign indicates whether the permutation is even or odd.

Question 32

What does it mean for two permutations to be conjugate?

Answer 32

Two permutations are conjugate if one can be transformed into the other by a change of basis in the symmetric group.

Question 33

What is a representation of a permutation group?

Answer 33

A representation is a homomorphism from the group to the general linear group of some vector space.

Question 34

Fill in the blank: A permutation can be represented by a _____ on the set of its elements.

Answer 34

matrix

Question 35

What is the significance of the orbit-stabilizer theorem?

Answer 35

It relates the size of the orbit of an element to the size of its stabilizer.

Question 36

True or False: The alternating group A_n is simple for n ≥ 5.

Answer 36

True

Question 37

What is the relationship between the symmetric group and combinatorial objects?

Answer 37

The symmetric group acts on combinatorial objects such as permutations and combinations.

Question 38

Name one application of permutation groups in modern mathematics.

Answer 38

One application is in the study of algebraic structures, such as Galois theory.