Quiz:Symmetric group:S3

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See symmetric group:S3. We take the symmetric group on the set \{ 1,2,3 \} of size three.

Elements

See element structure of symmetric group:S3 for full details.

Element orders and conjugacy class structure

Review the conjugacy class structure: [SHOW MORE]

1 What is the number of non-identity elements of the symmetric group of degree three?

2
3
4
5
6

2 How many elements are there of order exactly three in the symmetric group of degree three?

2
3
4
5
6

3 Which of the following is a correct description of the conjugacy class structure of the symmetric group of degree three?

Conjugacy class of size 1 and order (of elements in the conjugacy class) 1, conjugacy class of size 2 and order (of elements in the conjugacy class) 2, conjugacy class of size 3 and order (of elements in the conjugacy class) 3
Conjugacy class of size 1 and order (of elements in the conjugacy class) 1, conjugacy class of size 2 and order (of elements in the conjugacy class) 3, conjugacy class of size 3 and order (of elements in the conjugacy class) 2
Conjugacy class of size 1 and order (of elements in the conjugacy class) 2, conjugacy class of size 2 and order (of elements in the conjugacy class) 1, conjugacy class of size 3 and order (of elements in the conjugacy class) 3
Conjugacy class of size 1 and order (of elements in the conjugacy class) 2, conjugacy class of size 2 and order (of elements in the conjugacy class) 3, conjugacy class of size 3 and order (of elements in the conjugacy class) 1
Conjugacy class of size 1 and order (of elements in the conjugacy class) 3, conjugacy class of size 2 and order (of elements in the conjugacy class) 1, conjugacy class of size 3 and order (of elements in the conjugacy class) 2
Conjugacy class of size 1 and order (of elements in the conjugacy class) 3, conjugacy class of size 2 and order (of elements in the conjugacy class) 2, conjugacy class of size 3 and order (of elements in the conjugacy class) 1

Multiplication, conjugacy and generating sets

Review the multiplication table in cycle decomposition notation: [SHOW MORE]


Review the multiplication table in one-line notation: [SHOW MORE]


1 What can we say about the order of the product of two distinct elements, each of order exactly two, in the symmetric group of degree three?

The product must be the identity element
The product must have order two
The product can have order either 1 or 2
The product must have order three
The product can have order either 1 or 3

2 What can we say about the order of the product of two distinct elements, each of order exactly three, in the symmetric group of degree three?

The product must be the identity element
The product must have order two
The product can have order either 1 or 2
The product must have order three
The product can have order either 1 or 3

3 Which of the following is false in the symmetric group of degree three?

Any two elements of the same order are conjugate
Every element is conjugate to its inverse
Any two elements generating the same cyclic subgroup are conjugate
Any two elements that together generate the whole group are conjugate
None of the above, i.e., they are all true

Conjugation and commutator operations

Review the conjugation operation: [SHOW MORE]


Review the commutator operation: [SHOW MORE]

1 Suppose g and h are distinct elements of order two in the symmetric group of order three. What can we say about ghg^{-1} (this is a conjugate of h by g)?

It equals g
It equals h
It equals an element of order two that is neither g nor h
It is an element of order three
It is the identity element

2 Suppose g and h are distinct elements of order two in the symmetric group of order three. What can we say about the commutator ghg^{-1}h^{-1}?

It equals g
It equals h
It equals an element of order two that is neither g nor h
It is an element of order three
It is the identity element

3 Suppose g and h are distinct elements of order three in the symmetric group of order three. What can we say about ghg^{-1} (this is a conjugate of h by g)?

It equals g
It equals h
It equals an element of order two
It is the identity element

4 Suppose g and h are distinct elements of order three in the symmetric group of order three. What can we say about the commutator ghg^{-1}h^{-1}?

It equals g
It equals h
It is an element of order two
It is the identity element

5 Suppose g is an element of order two and h is an element of order three in the symmetric group of order three. What are the orders of the elements ghg^{-1} and hgh^{-1} respectively?

1 and 1
2 and 3
3 and 2
2 and 2
3 and 3

Subgroups

See subgroup structure of symmetric group:S3 for background information.

Basic stuff

Summary table on the structure of subgroups: [SHOW MORE]


View the lattice of subgroups as a picture: [SHOW MORE]

1 What are the possible orders of subgroups of the symmetric group of degree three?

1 and 6 only
1, 2, and 6 only
1, 3, and 6 only
1, 2, 3, and 6 only
1, 4, and 6 only
1, 2, 3, 4, 5, and 6 only

2 What are the possible orders of normal subgroups of the symmetric group of degree three?

1 and 6 only
1, 2, and 6 only
1, 3, and 6 only
1, 2, 3, and 6 only
1, 4, and 6 only
1, 2, 3, 4, 5, and 6 only

3 What are the possible orders of quotient groups of the symmetric group of degree three?

1 and 6 only
1, 2, and 6 only
1, 3, and 6 only
1, 2, 3, and 6 only
1, 4, and 6 only
1, 2, 3, 4, 5, and 6 only

Linear representations

For background, see linear representation theory of symmetric group:S3.

View overall summary: [SHOW MORE]


View representations summary: [SHOW MORE]


View character table: [SHOW MORE]

Basic stuff

Up to equivalence, there are three irreducible representations of symmetric group:S3 in characteristic zero: the one-dimensional trivial representation, the one-dimensional sign representation (that sends every permutation to its sign), and the standard representation of symmetric group:S3, a two-dimensional representation.

1 Which of the irreducible representations is realized over the field of rational numbers?

trivial representation only
trivial representation and sign representation only
all three representations

2 Which of the irreducible representations can be realized using orthogonal matrices (i.e., matrices in the orthogonal group for the standard dot product) over the field of rational numbers?

trivial representation only
trivial representation and sign representation only
all three representations

3 The tensor product of the standard representation and the sign representation is a representation of the symmetric group of degree three. What representation is it?

the standard representation
the sum of the trivial and the sign representation
the sum of two copies of the sign representation
the sum of two copies of the trivial representation

4 The symmetric group of degree three can also be viewed as a dihedral group of degree three and order six, acting on a set of size three. The 3-cycles become rotations and the transpositions become reflections. This defines a two-dimensional representation over the real numbers. Which of these is the two-dimensional representation?

the standard representation
the sum of the trivial and the sign representation
the sum of two copies of the sign representation
the sum of two copies of the trivial representation