Quiz:Element structure of symmetric group:S3

From Groupprops
Jump to: navigation, search

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