Quiz:Element structure of symmetric group:S5

See element structure of symmetric group:S5 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	Representative element with the cycle type	Size of conjugacy class	Formula calculating size	Even or odd? If even, splits? If splits, real in alternating group?	Element order	Formula calcuating element order
1 + 1 + 1 + 1 + 1	1 (5 times)	five fixed points	$()$ -- the identity element	1	$\!{\frac {5!}{(1)^{5}(5!)}}$	even; no	1	$\operatorname {lcm} \{1\}$
2 + 1 + 1 + 1	2 (1 time), 1 (3 times)	transposition: one 2-cycle, three fixed point	$(1,2)$	10	$\!{\frac {5!}{[(2)^{1}(1!)][(1)^{3}(3!)]}}$ or $\!{\frac {5!}{(2)(1)^{3}(3!)}}$, also ${\binom {5}{2}}$ in this case	odd	2	$\operatorname {lcm} \{2,1\}$
3 + 1 + 1	3 (1 time), 1 (2 times)	one 3-cycle, two fixed points	$(1,2,3)$	20	$\!{\frac {5!}{[(3)^{1}(1!)][(1)^{2}(2!)]}}$ or $\!{\frac {5!}{(3)(1)^{2}(2!)}}$	even; no	3	$\operatorname {lcm} \{3,1\}$
2 + 2 + 1	2 (2 times), 1 (1 time)	double transposition: two 2-cycles, one fixed point	$(1,2)(3,4)$	15	${\frac {5!}{[2^{2}(2!)][1^{1}(1!)]}}$ or $\!{\frac {5!}{2^{2}(2!)(1)}}$	even; no	2	$\operatorname {lcm} \{2,1\}$
4 + 1	4 (1 time), 1 (1 time)	one 4-cycle, one fixed point	$(1,2,3,4)$	30	$\!{\frac {5!}{[4^{1}(1!)][1^{1}(1!)]}}$ or$\!{\frac {5!}{(4)(1)}}$	odd	4	$\operatorname {lcm} \{4,1\}$
3 + 2	3 (1 time), 2 (1 time)	one 3-cycle, one 2-cycle	$(1,2,3)(4,5)$	20	$\!{\frac {5!}{[3^{1}(1!)][2^{1}(1!)]}}$ or $\!{\frac {5!}{(3)(2)}}$	odd	6	$\operatorname {lcm} \{3,2\}$
5	5 (1 time)	one 5-cycle	$(1,2,3,4,5)$	24	${\frac {5!}{5^{1}(1!)}}$ or $\!{\frac {5!}{5}}$	even; yes; yes	5	$\operatorname {lcm} \{5\}$
Total (7 rows, 7 being the number of unordered integer partitions of 5)	--	--	--	120 (equals order of the group)	--	odd: 60 (3 classes) even;no: 36 (3 classes) even;yes;yes: 24 (1 class)	[SHOW MORE] order 1: 1 (1 class) order 2: 25 (2 classes) order 3: 20 (1 class) order 4: 30 (1 class) order 5: 24 (1 class) order 6: 20 (1 class)

1 What is the maximum among the orders of elements of the symmetric group of degree five?

	5
	6
	8
	10
	12

2 What is the exponent of the symmetric group of degree five, i.e., the least common multiple of the orders of all elements of this group?

	10
	20
	30
	60
	120