Symmetric group:S5

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Definition

The symmetric group $S_5$ is defined in the following equivalent ways:

1. It is the group of all permutations on a set of five elements, i.e., it is the symmetric group of degree five. In particular, it is a symmetric group of prime degree and symmetric group of prime power degree. With this interpretation, it is denoted $S_5$ or $\operatorname{Sym}(5)$.
2. It is the projective general linear group of degree two over the field of five elements, i.e., $PGL(2,5)$.
3. It is the projective semilinear group of degree two over the field of four elements, i.e., $P\Gamma L(2,4)$.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 120 groups with same order As $\! S_n, n = 5:$ $\! n! = 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$

As $\! PGL(2,q), q = 5$ (see order formulas for linear groups of degree two): $\! q^3 - q = q(q-1)(q+1) = 5^3 - 5 = 5(4)(6) = 120$
As $\! P\Gamma L(2,q), q = p^r, q = 4, p = 2, r = 2$ (see order formulas for linear groups of degree two): $\! r(q^3 - q) = 2(4^3 - 4) = 2(60) = 120$.
exponent 60 groups with same order and exponent | groups with same exponent As $\! S_n, n = 5:$ $\! \operatorname{lcm} \{ 1,2,\dots,n \} = \operatorname{lcm} \{1,2,3,4,5 \} = 60$

As $\! PGL(2,q), q = p = 5:$ (where $\! p$ is the underlying prime for $\! q$) $\! p(q^2 - 1)/2 = 5 \cdot (5^2 - 1)/2 = 60$
derived length -- not a solvable group.
nilpotency class -- not a nilpotent group.
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length Frattini-free group: intersection of maximal subgroups is trivial.
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set $\! (1,2), (1,2,3,4,5)$; see also symmetric group on a finite set is 2-generated
subgroup rank of a group 2 groups with same order and subgroup rank of a group | groups with same subgroup rank of a group
max-length of a group 5 groups with same order and max-length of a group | groups with same max-length of a group
number of subgroups 156 groups with same order and number of subgroups | groups with same number of subgroups
number of conjugacy classes 7 groups with same order and number of conjugacy classes | groups with same number of conjugacy classes As $\! S_k, k = 5$: the number of conjugacy classes is $\! p(k) = p(5) = 7$, where $p$ is the number of unordered integer partitions. See also cycle type determines conjugacy class

As $\! PGL(2,q), q = 5:$ $\! q + 2 = 5 + 2 = 7$
number of conjugacy classes of subgroups 19 groups with same order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups

Group properties

COMPARE AND CONTRAST: Want to know more about how this group compares with symmetric groups of other degrees? Read contrasting symmetric groups of various degrees.

Basic properties

Property Satisfied? Explanation Comment
abelian group No $(1,2)$, $(1,3)$ don't commute $S_n$ is non-abelian, $n \ge 3$.
nilpotent group No Centerless: The center is trivial $S_n$ is non-nilpotent, $n \ge 3$.
metacyclic group No No cyclic normal subgroup $S_n$ is not metacyclic, $n \ge 4$.
supersolvable group No No cyclic normal subgroup $S_n$ is not supersolvable, $n \ge 4$.
solvable group No The subgroup $A_5$ is simple non-abelian $A_n$ is simple and hence $S_n$ not solvable, $n \ge 5$.
simple non-abelian group No has a proper nontrivial normal subgroup A5 in S5.
almost simple group Yes It contains a centralizer-free simple normal subgroup, namely A5 in S5. symmetric groups are almost simple for degree 5 or higher.
perfect group No Its derived subgroup is A5 in S5 and abelianization is cyclic group:Z2.
quasisimple group No Follows from not being perfect.
centerless group Yes symmetric groups are centerless symmetric groups are centerless for degree other than two.
complete group Yes Centerless and every automorphism's inner Symmetric groups are complete except the ones of degree $2,6$.

Other properties

Property Satisfied? Explanation Comment
T-group Yes
HN-group Yes
monolithic group Yes Monolith is the alternating group All symmetric groups are monolithic; $n=4$ is the only case the monolith is not the alternating group.
one-headed group Yes The alternating group is the unique maximal normal subgroup True for all $n > 1$.

Elements

Further information: element structure of symmetric group:S5

Upto conjugacy

For convenience, we take the underlying set to be $\{ 1,2,3,4,5 \}$.

There are seven conjugacy classes, corresponding to the unordered integer partitions of $5$ (for more information, refer cycle type determines conjugacy class). We use the notation of the cycle decomposition for permutations:

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)

Upto automorphism

$S_5$ is a complete group: in particular, every automorphism of the group is inner. Thus, the equivalence classes under automorphisms are the same as the conjugacy classes. In fact, $S_n$ is complete for $n \ne 2, 6$. See symmetric groups on finite sets are complete.

Endomorphisms

Further information: endomorphism structure of symmetric group:S5

Summary information

Construct Value Order Second part of GAP ID (if group)
endomorphism monoid  ? 146 --
automorphism group symmetric group:S5 120 34
inner automorphism group symmetric group:S5 120 34
extended automorphism group direct product of S5 and Z2 240 189

Automorphisms

Since $S_5$ is a complete group, it is isomorphic to its automorphism group, where each element of $S_5$ acts on $S_5$ by conjugation. Further information: symmetric groups on finite sets are complete

Subgroups

Further information: Subgroup structure of symmetric group:S5

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 --

Linear representation theory

Further information: linear representation theory of symmetric group:S5

Summary

Item Value
Degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,1,4,4,5,5,6
maximum: 6, lcm: 60, number: 7, sum of squares: 120
Schur index values of irreducible representations 1,1,1,1,1,1,1
maximum: 1, lcm: 1
Smallest ring of realization for all irreducible representations (characteristic zero) $\mathbb{Z}$ -- ring of integers
Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) $\mathbb{Q}$ -- hence it is a rational representation group
Criterion for a field to be a splitting field Any field of characteristic not equal to 2,3, or 5.
Smallest size splitting field field:F7, i.e., the field of 7 elements.

Character table

Representation/conjugacy class representative and size $()$ (size 1) $(1,2)$ (size 10) $(1,2)(3,4)$ (size 15) $(1,2,3)$ (size 20) $(1,2,3)(4,5)$ (size 20) $(1,2,3,4,5)$ (size 24) $(1,2,3,4)$ (size 30)
trivial representation 1 1 1 1 1 1 1
sign representation 1 -1 1 1 -1 1 -1
standard representation 4 2 0 1 -1 -1 0
product of standard and sign representation 4 -2 0 1 1 -1 0
irreducible five-dimensional representation 5 1 1 -1 1 0 -1
irreducible five-dimensional representation 5 -1 1 -1 -1 0 1
exterior square of standard representation 6 0 -2 0 0 1 0

GAP implementation

Group ID

This finite group has order 120 and has ID 34 among the groups of order 120 in GAP's SmallGroup library. For context, there are 47 groups of order 120. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(120,34)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(120,34);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [120,34]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

Description Functions used
SymmetricGroup(5) SymmetricGroup
PGL(2,5) PGL