Quiz:Symmetric group:S5

See symmetric group:S5. We take the symmetric group on the set $\! \{ 1,2,3,4,5 \}$ of size five.

Elements

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

Subgroups

See subgroup structure of symmetric group:S5 for background information and more details.

Basic stuff

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

Quick summary

Item	Value
Number of subgroups	156 Compared with $S_n, n = 1,2,3,4,5,6,7,\dots$: 1,2,6,30,156,1455,11300, 151221
Number of conjugacy classes of subgroups	19 Compared with $S_n$, $n = 1,2,3,4,5,6,7,\dots$: 1,2,4,11,19,56,96,296,554,1593
Number of automorphism classes of subgroups	19 Compared with $S_n$, $n = 1,2,3,4,5,6,7,\dots$: 1,2,4,11,19,37,96,296,554,1593
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems	2-Sylow: dihedral group:D8 (order 8), Sylow number is 15, fusion system is non-inner non-simple fusion system for dihedral group:D8 3-Sylow: cyclic group:Z3, Sylow number is 10, fusion system is non-inner fusion system for cyclic group:Z3 5-Sylow: Z5 in S5, Sylow number is 6, fusion system is universal fusion system for cyclic group:Z5
Hall subgroups	$\{ 2,3 \}$-Hall subgroup: S4 in S5 (order 24) No $\{ 2,5 \}$-Hall subgroup or $\{ 3,5 \}$-Hall subgroup
maximal subgroups	maximal subgroups have orders 12 (direct product of S3 and S2 in S5), 20 (GA(1,5) in S5), 24 (S4 in S5), 60 (A5 in S5)
normal subgroups	There are three normal subgroups: the whole group, A5 in S5, and the trivial subgroup.

Table classifying subgroups up to automorphisms

Note that the only normal subgroups are the trivial subgroup, the whole group, and A5 in S5, so we do not waste a column on specifying whether the subgroup is normal and on the quotient group.

TABLE SORTING AND INTERPRETATION: Note that the subgroups in the table below are sorted based on the powers of the prime divisors of the order, first covering the smallest prime in ascending order of powers, then powers of the next prime, then products of powers of the first two primes, then the third prime, then products of powers of the first and third, second and third, and all three primes. The rationale is to cluster together subgroups with similar prime powers in their order. The subgroups are not sorted by the magnitude of the order. To sort that way, click the sorting button for the order column. Similarly you can sort by index or by number of subgroups of the automorphism class.

Automorphism class of subgroups	Representative subgroup (full list if small, generating set if large)	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (= 1 iff automorph-conjugate subgroup)	Size of each conjugacy class (= 1 iff normal subgroup)	Total number of subgroups (= 1 iff characteristic subgroup)	Note
trivial subgroup	$()$	trivial group	1	120	1	1	1	trivial
S2 in S5	$\{ (), (1,2) \}$	cyclic group:Z2	2	60	1	10	10
subgroup generated by double transposition in S5	$\{ (), (1,2)(3,4) \}$	cyclic group:Z2	2	60	1	15	15
subgroup generated by pair of disjoint transpositions in S5	$\{ (), (1,2), (3,4), (1,2)(3,4) \}$	Klein four-group	4	30	1	15	15
subgroup generated by double transpositions on four elements in S5	$\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$	Klein four-group	4	30	1	5	5
Z4 in S5	$\{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) \}$	cyclic group:Z4	4	30	1	15	15
D8 in S5	$\langle (1,2,3,4), (1,3) \rangle$	dihedral group:D8	8	15	1	15	15	2-Sylow
Z3 in S5	$\{ (), (1,2,3), (1,3,2) \}$	cyclic group:Z3	3	40	1	10	10	3-Sylow
Z6 in S5	$\langle (1,2,3), (4,5) \rangle$	cyclic group:Z6	6	20	1	10	10
S3 in S5	$\langle (1,2,3), (1,2) \rangle$	symmetric group:S3	6	20	1	10	10
twisted S3 in S5	$\langle (1,2,3), (1,2)(4,5) \rangle$	symmetric group:S3	6	20	1	10	10
direct product of S3 and S2 in S5	$\langle (1,2,3), (1,2), (4,5) \rangle$	direct product of S3 and Z2	12	10	1	10	10	3-Sylow normalizer
A4 in S5	$\langle (1,2)(3,4), (1,2,3) \rangle$	alternating group:A4	12	10	1	5	5
S4 in S5	$\langle (1,2,3,4), (1,2) \rangle$	symmetric group:S4	24	5	1	5	5	(2,3)-Hall
Z5 in S5	$\langle (1,2,3,4,5) \rangle$	cyclic group:Z5	5	24	1	6	6	5-Sylow
D10 in S5	$\langle (1,2,3,4,5), (2,5)(3,4) \rangle$	dihedral group:D10	10	12	1	6	6
GA(1,5) in S5	$\langle (1,2,3,4,5), (2,3,5,4) \rangle$	general affine group:GA(1,5)	20	6	1	6	6
A5 in S5	$\langle (1,2,3,4,5), (1,2,3)\rangle$	alternating group:A5	60	2	1	1	1	only proper nontrivial normal subgroup, quotient is cyclic group:Z2
whole group	$\langle (1,2,3,4,5), (1,2) \rangle$	symmetric group:S5	120	1	1	1	1
Total (19 rows)	--	--	--	--	19	--	156	--

1 For which of the following divisors of 120 does there not exist a subgroup of the symmetric group of degree five with that as order?

	12
	20
	24
	30
	60

2 What is the smallest possible order of a group that is not isomorphic to any subgroup of the symmetric group of degree five?

	5
	6
	7
	8
	9

3 What is the smallest possible order of a group that is isomorphic to subgroups in two distinct conjugacy classes of subgroups in the symmetric group of degree five?

	2
	3
	4
	5
	6