Quiz:Element structure of symmetric group:S4

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

Element orders and conjugacy class structure

Review the conjugacy class structure: [SHOW MORE]

Partition	Partition in grouped form	Verbal description of cycle type	Elements with the cycle type	Size of conjugacy class	Formula for size	Even or odd? If even, splits? If splits, real in alternating group?	Element order	Formula calculating element order
1 + 1 + 1 + 1	1 (4 times)	four cycles of size one each, i.e., four fixed points	$()$ -- the identity element	1	$\! \frac{4!}{(1)^4(4!)}$	even; no	1	$\operatorname{lcm}\{ 1,1,1,1 \}$
2 + 1 + 1	2 (1 time), 1 (2 times)	one transposition (cycle of size two), two fixed points	$(1,2)$, $(1,3)$, $(1,4)$, $(2,3)$, $(2,4)$, $(3,4)$	6	$\! \frac{4!}{[(2)^1(1!)][(1)^2(2!)]}$, also $\binom{4}{2}$	odd	2	$\operatorname{lcm}\{2,1,1 \}$
2 + 2	2 (2 times)	double transposition: two cycles of size two	$(1,2)(3,4)$, $(1,3)(2,4)$, $(1,4)(2,3)$	3	$\! \frac{4!}{(2)^2(2!)}$	even; no	2	$\operatorname{lcm}\{2,2 \}$
3 + 1	3 (1 time), 1 (1 time)	one 3-cycle, one fixed point	$(1,2,3)$, $(1,3,2)$, $(2,3,4)$, $(2,4,3)$, $(3,4,1)$, $(3,1,4)$, $(4,1,2)$, $(4,2,1)$	8	$\! \frac{4!}{[(3)^1(1!)][(1)^1(1!)]}$ or $\! \frac{4!}{(3)(1)}$	even; yes; no	3	$\operatorname{lcm}\{3,1 \}$
4	4 (1 time)	one 4-cycle, no fixed points	$(1,2,3,4)$, $(1,2,4,3)$, $(1,3,2,4)$, $(1,3,4,2)$, $(1,4,2,3)$, $(1,4,3,2)$	6	$\! \frac{4!}{(4)^1(1!)}$ or $\frac{4!}{4}$	odd	4	$\operatorname{lcm} \{ 4 \}$
Total (5 rows, 5 being the number of unordered integer partitions of 4)	--	--	--	24 (equals 4!, the order of the whole group)	--	odd: 12 (2 classes) even; no: 4 (2 classes) even; yes; no: 8 (1 class)	order 1: 1 (1 class) order 2: 9 (2 classes) order 3: 8 (1 class) order 4: 6 (1 class)	--

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?

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