Subgroup structure of symmetric group:S5: Difference between revisions

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

Revision as of 23:59, 8 January 2012

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This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S5.
View subgroup structure of particular groups | View other specific information about symmetric group:S5

The symmetric group of degree five has many subgroups. We'll take the five letters as {1,2,3,4,5}. The group has order 120.

Note that since S5 is a complete group, every automorphism of it is inner, so the classification of subgroups upto conjugacy is the same as the classification of subgroups upto automorphism. In other words, every subgroup is an automorph-conjugate subgroup.

Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate

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.

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 Size of each conjugacy class Total number of subgroups 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 ⟨(1,2,3,4),(1,3)⟩ 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 ⟨(1,2,3),(4,5)⟩ cyclic group:Z6 6 20 1 10 10
S3 in S5 ⟨(1,2,3),(1,2)⟩ symmetric group:S3 6 20 1 10 10
twisted S3 in S5 ⟨(1,2,3),(1,2)(4,5)⟩ symmetric group:S3 6 20 1 10 10
direct product of S3 and S2 in S5 ⟨(1,2,3),(1,2),(4,5)⟩ direct product of S3 and Z2 12 10 1 10 10 3-Sylow normalizer
A4 in S5 ⟨(1,2)(3,4),(1,2,3)⟩ alternating group:A4 12 10 1 5 5
S4 in S5 ⟨(1,2,3,4),(1,2)⟩ symmetric group:S4 24 5 1 5 5 (2,3)-Hall
Z5 in S5 ⟨(1,2,3,4,5)⟩ cyclic group:Z5 5 24 1 6 6 5-Sylow
D10 in S5 ⟨(1,2,3,4,5),(2,5)(3,4)⟩ dihedral group:D10 10 12 1 6 6
GA(1,5) in S5 ⟨(1,2,3,4,5),(2,3,5,4)⟩ general affine group:GA(1,5) 20 6 1 6 6
A5 in S5 ⟨(1,2,3,4,5),(1,2,3)⟩ alternating group:A5 60 2 1 1 1 only proper nontrivial normal subgroup, quotient is cyclic group:Z2
whole group ⟨(1,2,3,4,5),(1,2)⟩ symmetric group:S5 120 1 1 1 1
Total (19 rows) -- -- -- -- 19 -- 156 --

Table classifying isomorphism types of subgroups

Group name Order Second part of GAP ID (first part is order) Occurrences as subgroup Conjugacy classes of occurrence as subgroup Automorphism classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
trivial group 1 1 1 1 1 1 1
cyclic group:Z2 2 1 25 2 2 0 0
cyclic group:Z3 3 1 10 1 1 0 0
cyclic group:Z4 4 1 15 1 1 0 0
Klein four-group 4 2 20 2 2 0 0
cyclic group:Z5 5 1 6 1 1 0 0
symmetric group:S3 6 1 20 2 2 0 0
cyclic group:Z6 6 2 10 1 1 0 0
dihedral group:D8 8 3 15 1 1 0 0
dihedral group:D10 10 1 6 1 1 0 0
alternating group:A4 12 3 5 1 1 0 0
direct product of S3 and Z2 12 4 10 1 1 0 0
general affine group:GA(1,5) 20 3 6 1 1 0 0
symmetric group:S4 24 12 5 1 1 0 0
alternating group:A5 60 5 1 1 1 1 1
symmetric group:S5 120 34 1 1 1 1 1
Total -- -- 156 19 19 3 3

Table listing number of subgroups by order

Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order pr is congruent to 1 modulo p.

Group order Occurrences as subgroup Conjugacy classes of occurrence as subgroup Automorphism classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
1 1 1 1 1 1
2 25 2 2 0 0
3 10 1 1 0 0
4 35 3 3 0 0
5 6 1 1 0 0
6 30 3 3 0 0
8 15 1 1 0 0
10 6 1 1 0 0
12 15 2 2 0 0
20 6 1 1 0 0
24 5 1 1 0 0
60 1 1 1 1 1
120 1 1 1 1 1
Total 156 19 19 3 3
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