Subgroup structure of symmetric group:S5: Difference between revisions

Revision as of 17:13, 22 June 2011

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 $S_{5}$ 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

Table classifying subgroups up to automorphisms

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
trivial subgroup	$()$	trivial group	1	120	1	1	1
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
Z3 in S5	$\{(),(1,2,3),(1,3,2)\}$	cyclic group:Z3	3	40	1	10	10
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
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
Z5 in S5	$\langle (1,2,3,4,5)\rangle$	cyclic group:Z5	5	24	1	6	6
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
whole group	$\langle (1,2,3,4,5),(1,2)\rangle$	symmetric group:S5	120	1	1	1	1
Total	--	--	--	--	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 $p^{r}$ 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

@@ Line 4: / Line 4: @@
 connective = of}}
-The [[symmetric group:S5|symmetric group of degree five]] has many subgroups. We'll take the five letters as <math>\{ 1,2,3,4,5\}</math>.
+The [[symmetric group:S5|symmetric group of degree five]] has many subgroups. We'll take the five letters as <math>\{ 1,2,3,4,5\}</math>. The group has order 120.
 Note that since <math>S_5</math> 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]].
-# The trivial subgroup. Isomorphic to [[subgroup::trivial group]]. (1)
-# The two-element subgroup generated by a transposition, such as <math>(1,2)</math>. Isomorphic to [[cyclic group:Z2|cyclic group of order two]]. (10).
-# The two-element subgroup generated by a double transposition, such as <math>(1,2)(3,4)</math>. Isomorphic to [[cyclic group:Z2|cyclic group of order two]]. (15)
-# The four-element subgroup spanned by two disjoint transpositions, such as <math>\langle (1,2) , (3,4) \rangle</math>. Isomorphic to [[Klein four-group]]. (15)
-# The four-element subgroup containing the identity and three double transpositions on a subset of size four. Isomorphic to [[Klein four-group]]. (5)
-# The four-element subgroup spanned by a 4-cycle. Isomorphic to [[cyclic group:Z4|cyclic group of order four]]. (15)
-# The eight-element subgroup spanned by a 4-cycle and a transposition that conjugates this cycle to its inverse. Isomorphic to [[dihedral group:D8|dihedral group of order eight]]. (15)
-# The three-element subgroup spanned by a three-cycle. Isomorphic to [[cyclic group:Z3|cyclic group of order three]]. (10)
-# The six-element spanned by a 3-cycle and a transposition disjoint from it. Isomorphic to [[cyclic group:Z6|cyclic group of order six]]. (10)
-# The six-element subgroup spanned by all permutations on a subset of size three. Isomorphic to [[symmetric group:S3|symmetric group on three elements]]. (10)
-# The six-element subgroup obtained by taking permutations on a particular subset of size three, and multiplying this by a transposition on the remaining two elements if the permutation is odd. Isomorphic to [[symmetric group:S3|symmetric group on three letters]]. (10)
-# The twelve-element subgroup generated by the symmetric group on three letters and the symmetric group on the remaining two letters. Isomorphic to [[direct product of S3 and Z2]]. (10)
-# The twelve-element subgroup obtained as the alternating group on four letters. Isomorphic to [[alternating group:A4|alternating group on four letters]]. (5)
-# The 24-element subgroup obtained as the symmetric group on four letters. Isomorphic to [[symmetric group:S4|symmetric group on four letters]]. (5)
-# The five-element subgroup generated by a five-cycle. Isomorphic to [[cyclic group:Z5|cyclic group on five letters]]. (6)
-# The ten-element subgroup generated by a five-cycle and a double transposition that conjugate it to its inverse. Isomorphic to [[dihedral group:D10|dihedral group of order ten]]. (6)
-# The twenty-element subgroup generated by a five-cycle and a four-cycle that conjugates it to its square. Isomorphic to [[general affine group:GA(1,5)]]. (6)
-# The alternating group on all five letters. Isomorphic to [[subgroup::alternating group:A5|alternating group on five letters]]. (1)
-# The whole group. (1)
 ==Tables for quick information==
@@ Line 33: / Line 14: @@
 {| class="sortable" border="1"
-! Automorphism class of subgroups !! Representative subgroup (full list if small, generating set if large) !! Isomorphism class !! 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
 |-
-| trivial subgroup || <math>()</math> || [[trivial group]] || 1 || 1 || 1
+| trivial subgroup || <math>()</math> || [[trivial group]] || 1 || 120 || 1 || 1 || 1
 |-
-| [[S2 in S5]] || <math>\{ (), (1,2) \}</math> || [[cyclic group:Z2]] || 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]] || 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]] || 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]] || 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]] || 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]] || 1 || 15 || 15
+| [[D8 in S5]] || <math>\langle (1,2,3,4), (1,3) \rangle</math> || [[dihedral group:D8]] || 8 || 15 || 1 || 15 || 15
 |-
-| [[Z3 in S5]] || <math>\{ (), (1,2,3), (1,3,2) \}</math> || [[cyclic group:Z3]] || 1 || 10 || 10
+| [[Z3 in S5]] || <math>\{ (), (1,2,3), (1,3,2) \}</math> || [[cyclic group:Z3]] || 3 || 40 || 1 || 10 || 10
 |-
-| [[Z6 in S5]] || <math>\langle (1,2,3), (4,5) \rangle</math> || [[cyclic group:Z6]] || 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]] || 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]] || 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]] || 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
 |-
-| [[A4 in S5]] || <math>\langle (1,2)(3,4), (1,2,3) \rangle</math> || [[alternating group:A4]] || 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]] || 1 || 5 || 5
+| [[S4 in S5]] || <math>\langle (1,2,3,4), (1,2) \rangle</math> || [[symmetric group:S4]] || 24 || 5 || 1 || 5 || 5
 |-
-| [[Z5 in S5]] || <math>\langle (1,2,3,4,5) \rangle</math> || [[cyclic group:Z5]] || 1 || 6 || 6
+| [[Z5 in S5]] || <math>\langle (1,2,3,4,5) \rangle</math> || [[cyclic group:Z5]] || 5 || 24 || 1 || 6 || 6
 |-
-| [[D10 in S5]] || <math>\langle (1,2,3,4,5), (2,5)(3,4) \rangle</math> || [[dihedral group:D10]] || 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)]] || 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]] || 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
 |-
-| whole group || <math>\langle (1,2,3,4,5), (1,2) \rangle</math> || [[symmetric group:S5]] || 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 || -- || 156
 |}