# Quiz:Symmetric group:S4

See symmetric group:S4. We take the symmetric group on the set $\! \{1,2,3,4\}$ of size four.

## Elements

See element structure of symmetric group:S4 for full details.

### Element orders and conjugacy class structure

Review the conjugacy class structure: [SHOW MORE]

1 What are the possible orders of elements of the symmetric group of degree four?

 1,2,3,4 only 1,2,3,4,6 only 1,2,3,4,8 only 1,2,3,4,6,8 only 1,2,3,4,6,8,12 only

2 For which of the following orders of elements is there more than one conjugacy class of elements with that order in the symmetric group of degree four?

 1 2 3 4 6 8 12

3 Which of the following is a representative of the conjugacy class in the symmetric group of degree four that has the largest size?

 $(1,2)$ $(1,2,3)$ $(1,2,3,4)$ $(1,2)(3,4)$

## Subgroups

See subgroup structure of symmetric group:S4 for background information.

### Basic stuff

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

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

 1,2,6,24 only 1,3,8,24 only 1,4,12,24 only 1,2,3,4,12,24 only 1,2,3,4,6,12,24 only 1,2,3,4,8,12,24 only 1,2,3,4,6,8,12,24 only

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

 1,2,6,24 only 1,3,8,24 only 1,4,12,24 only 1,2,3,4,12,24 only 1,2,3,4,6,12,24 only 1,2,3,4,8,12,24 only 1,2,3,4,6,8,12,24 only

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

 1,2,6,24 only 1,3,8,24 only 1,4,12,24 only 1,2,3,4,12,24 only 1,2,3,4,6,12,24 only 1,2,3,4,8,12,24 only 1,2,3,4,6,8,12,24 only

4 For which of the following orders of subgroups do there exist two distinct conjugacy classes of subgroups in the symmetric group of that order for which the subgroups are isomorphic as abstract groups?

 2 only 3 only 4 only 6 only 2 and 3 only 3 and 4 only 2 and 4 only 3 and 6 only