Element structure of symmetric group:S6: Difference between revisions

Latest revision as of 19:37, 11 February 2014

This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S6.
View element structure of particular groups | View other specific information about symmetric group:S6

This article describes the element structure of symmetric group:S6.

For convenience, we take the underlying set to be ${1, 2, 3, 4, 5, 6}$ .

This group is NOT isomorphic to projective general linear group:PGL(2,9). For proof of the non-isomorphism, see PGL(2,9) is not isomorphic to S6. For the element structure of that group, see element structure of projective general linear group:PGL(2,9).

Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Interpretation as symmetric group

FACTS TO CHECK AGAINST SPECIFICALLY FOR SYMMETRIC GROUPS AND ALTERNATING GROUPS:
Please read element structure of symmetric groups for a summary description.
Conjugacy class parametrization: cycle type determines conjugacy class (in symmetric group)
Conjugacy class sizes: conjugacy class size formula in symmetric group
Other facts: even permutation (definition) -- the alternating group is the set of even permutations | splitting criterion for conjugacy classes in the alternating group (from symmetric group)| criterion for element of alternating group to be real

For a symmetric group, cycle type determines conjugacy class, so the conjugacy classes are parametrized by the set of unordered integer partitions of the number 6.

Partition	Partition in grouped form	Verbal description of cycle type	Representative element	Size of conjugacy class	Formula for size	Even or odd? If even, splits? If splits, real in alternating group?	Element orders	Formula calculating element order
1 + 1 + 1 + 1 + 1 + 1	1 (6 times)	six fixed points	$()$ -- the identity element	1	$\frac{6!}{(1)^{6} (6!)}$	even; no	1	$l c m {1}$
2 + 1 + 1 + 1 + 1	2 (1 time), 1 (4 times)	transposition, four fixed points	$(1, 2)$	15	$\frac{6!}{(2) (1)^{4} (4!)}$	odd	2	$l c m {2, 1}$
3 + 1 + 1 + 1	3 (1 time), 1 (3 times)	one 3-cycle, three fixed points	$(1, 2, 3)$	40	$\frac{6!}{(3) (1)^{3} (3!)}$	even; no	3	$l c m {3, 1}$
4 + 1 + 1	4 (1 time), 1 (2 times)	one 4-cycle, two fixed points	$(1, 2, 3, 4)$	90	$\frac{6!}{(4) (1)^{2} (2!)}$	odd	4	$l c m {4, 1}$
2 + 2 + 1 + 1	2 (2 times), 1 (2 times)	double transposition: two 2-cycles, two fixed points	$(1, 2) (3, 4)$	45	$\frac{6!}{(2)^{2} (2!) (1)^{2} (2!)}$	even; no	2	$l c m {2, 1}$
5 + 1	5 (1 time), 1 (1 time)	one 5-cycle, one fixed point	$(1, 2, 3, 4, 5)$	144	$\frac{6!}{(5) (1)}$	even; yes; yes	5	$l c m {5, 1}$
3 + 2 + 1	3 (1 time), 2 (1 time), 1 (1 time)	one 3-cycle, one 2-cycle, one fixed point	$(1, 2, 3) (4, 5)$	120	$\frac{6!}{(3) (2) (1)}$	odd	6	$l c m {3, 2, 1}$
2 + 2 + 2	2 (3 times)	triple transposition	$(1, 2) (3, 4) (5, 6)$	15	$\frac{6!}{(2)^{3} (3!)}$	odd	2	$l c m {2}$
4 + 2	4 (1 time), 2 (1 time)	one 4-cycle, one 2-cycle	$(1, 2, 3, 4) (5, 6)$	90	$\frac{6!}{(4) (2)}$	even; no	4	$l c m {4, 2}$
3 + 3	3 (2 times)	two 3-cycles	$(1, 2, 3) (4, 5, 6)$	40	$\frac{6!}{(3)^{2} (2!)}$	even; no	3	$l c m {3}$
6	6 (1 time)	one 6-cycle	$(1, 2, 3, 4, 5, 6)$	120	$\frac{6!}{6}$	odd	6	$l c m {6}$
Total (11 rows = 11 conjugacy classes)	--	--	--	720 (equals order of the whole group)	--	odd: 360, 5 classes even;no: 216, 5 classes even;yes;yes: 144, 1 class	order 1: 1, order 2: 75, order 3: 80, order 4: 180, order 5: 144, order 6: 240	--

Automorphism class structure

Partitions for cycle types in one automorphism class	Representative elements for each	Size of each conjugacy class	Number of classes	Total size	Element orders
1 + 1 + 1 + 1 + 1 + 1	$()$	1	1	1	1
2 + 1 + 1 + 1 + 1, 2 + 2 + 2	$(1, 2)$ , $(1, 2) (3, 4) (5, 6)$	15	2	30	2
3 + 1 + 1 + 1, 3 + 3	$(1, 2, 3)$ , $(1, 2, 3) (4, 5, 6)$	40	2	80	3
4 + 1 + 1	$(1, 2, 3, 4)$	90	1	90	4
4 + 2	$(1, 2, 3, 4) (5, 6)$	90	1	90	4
5 + 1	$(1, 2, 3, 4, 5)$	144	1	144	5
3 + 2 + 1, 6	$(1, 2, 3) (4, 5)$ , $(1, 2, 3, 4, 5, 6)$	120	2	240	6
2 + 2 + 1 + 1	$(1, 2) (3, 4)$	45	1	45	2
Total (8 classes)	--	--	11	720	--

@@ Line 9: / Line 9: @@
 For convenience, we take the underlying set to be <math>\{ 1,2,3,4,5,6\}</math>.
+{{quotation|This group is '''NOT''' isomorphic to [[projective general linear group:PGL(2,9)]]. For proof of the non-isomorphism, see [[PGL(2,9) is not isomorphic to S6]]. For the element structure of that group, see [[element structure of projective general linear group:PGL(2,9)]].}}
 ==Conjugacy class structure==
+{{conjugacy class structure facts to check against}}
 ===Interpretation as symmetric group===
+{{symmetric group conjugacy class structure facts to check against}}
 For a [[symmetric group]], [[cycle type determines conjugacy class]], so the conjugacy classes are parametrized by the [[set of unordered integer partitions]] of the number 6.
@@ Line 19: / Line 25: @@
 {| class="sortable" border="1"
-! Partition !! Verbal description of cycle type !! Representative element !! Size of conjugacy class !! Formula for size !! Even or odd? If even, splits? If splits, real in alternating group? !! Element orders
+! Partition !! Partition in grouped form !! Verbal description of cycle type !! Representative element !! Size of conjugacy class !! [[conjugacy class size formula in symmetric group|Formula for size]] !! [[even permutation|Even or odd]]? [[splitting criterion for conjugacy classes in the alternating group|If even, splits]]? [[criterion for element of alternating group to be real|If splits, real in alternating group]]? !! Element orders !! [[order of permutation is lcm of cycle sizes|Formula calculating element order]]
+|-
+| 1 + 1 + 1 + 1 + 1 + 1 || 1 (6 times) || six fixed points || <math>()</math> -- the identity element || 1 || <math>\! \frac{6!}{(1)^6(6!)}</math> || even; no || 1 || <math>\operatorname{lcm}\{ 1 \}</math>
 |-
-| 1 + 1 + 1 + 1 + 1 + 1 || six fixed points || <math>()</math> -- the identity element || 1 || <math>\! \frac{6!}{(1)^6(6!)}</math> || even; no || 1
+| 2 + 1 + 1 + 1 + 1 || 2 (1 time), 1 (4 times) || transposition, four fixed points || <math>(1,2)</math> || 15 || <math>\! \frac{6!}{(2)(1)^4(4!)}</math> || odd || 2 || <math>\operatorname{lcm}\{ 2, 1 \}</math>
 |-
-| 2 + 1 + 1 + 1 + 1 || transposition, four fixed points || <math>(1,2)</math> || 15 || <math>\! \frac{6!}{(2)(1)^4(4!)}</math> || odd || 2
+| 3 + 1 + 1 + 1 || 3 (1 time), 1 (3 times) || one 3-cycle, three fixed points || <math>(1,2,3)</math> || 40 || <math>\! \frac{6!}{(3)(1)^3(3!)}</math> || even; no || 3 || <math>\operatorname{lcm}\{ 3,1 \}</math>
 |-
-| 3 + 1 + 1 + 1 || one 3-cycle, three fixed points || <math>(1,2,3)</math> || 40 || <math>\! \frac{6!}{(3)(1)^3(3!)}</math> || even; no || 3
+| 4 + 1 + 1  || 4 (1 time), 1 (2 times) || one 4-cycle, two fixed points || <math>(1,2,3,4)</math> || 90 || <math>\! \frac{6!}{(4)(1)^2(2!)}</math> || odd || 4 || <math>\operatorname{lcm}\{4,1 \}</math>
 |-
-| 4 + 1 + 1  || one 4-cycle, two fixed points || <math>(1,2,3,4)</math> || 90 || <math>\! \frac{6!}{(4)(1)^2(2!)}</math> || odd || 4
+| 2 + 2 + 1 + 1 || 2 (2 times), 1 (2 times) || double transposition: two 2-cycles, two fixed points || <math>(1,2)(3,4)</math> || 45 || <math>\frac{6!}{(2)^2(2!)(1)^2(2!)}</math> || even; no || 2 || <math>\operatorname{lcm}\{ 2, 1 \}</math>
 |-
-| 5 + 1 || one 5-cycle, one fixed point || <math>(1,2,3,4,5)</math> || 144 || <math>\! \frac{6!}{(5)(1)}</math> || even; yes; yes || 5
+| 5 + 1 || 5 (1 time), 1 (1 time) || one 5-cycle, one fixed point || <math>(1,2,3,4,5)</math> || 144 || <math>\! \frac{6!}{(5)(1)}</math> || even; yes; yes || 5 || <math>\operatorname{lcm} \{ 5, 1 \}</math>
 |-
-| 3 + 2 + 1 || one 3-cycle, one 2-cycle, one fixed point || <math>(1,2,3)(4,5)</math> || 120 || <math>\! \frac{6!}{(3)(2)(1)}</math> || odd || 6
+| 3 + 2 + 1 || 3 (1 time), 2 (1 time), 1 (1 time) || one 3-cycle, one 2-cycle, one fixed point || <math>(1,2,3)(4,5)</math> || 120 || <math>\! \frac{6!}{(3)(2)(1)}</math> || odd || 6 || <math>\operatorname{lcm}\{3,2,1 \}</math>
 |-
-| 2 + 2 + 1 + 1 || double transposition: two 2-cycles, two fixed points || <math>(1,2)(3,4)</math> || 45 || <math>\frac{6!}{(2)^2(2!)(1)^2(2!)}</math> || even; no || 2
+| 2 + 2 + 2 || 2 (3 times) || triple transposition || <math>(1,2)(3,4)(5,6)</math> || 15 || <math>\! \frac{6!}{(2)^3(3!)}</math> || odd || 2 || <math>\operatorname{lcm} \{ 2 \}</math>
 |-
-| 2 + 2 + 2 || triple transposition || <math>(1,2)(3,4)(5,6)</math> || 15 || <math>\! \frac{6!}{(2)^3(3!)}</math> || odd || 2
+| 4 + 2 || 4 (1 time), 2 (1 time) || one 4-cycle, one 2-cycle || <math>(1,2,3,4)(5,6)</math> || 90 || <math>\! \frac{6!}{(4)(2)}</math> || even; no || 4 || <math>\operatorname{lcm} \{ 4 , 2 \}</math>
 |-
-| 4 + 2 || one 4-cycle, one 2-cycle || <math>(1,2,3,4)(5,6)</math> || 90 || <math>\! \frac{6!}{(4)(2)}</math> || even; no || 4
+| 3 + 3 || 3 (2 times) || two 3-cycles || <math>(1,2,3)(4,5,6)</math> || 40 || <math>\! \frac{6!}{(3)^2(2!)}</math> || even; no || 3 || <math>\operatorname{lcm} \{ 3 \}</math>
 |-
-| 3 + 3 || two 3-cycles || <math>(1,2,3)(4,5,6)</math> || 40 || <math>\! \frac{6!}{(3)^2(2!)}</math> || even; no || 3
+| 6 || 6 (1 time) || one 6-cycle || <math>(1,2,3,4,5,6)</math> || 120 || <math>\! \frac{6!}{6}</math> || odd || 6 || <math>\operatorname{lcm} \{ 6 \}</math>
 |-
-| 6 || one 6-cycle || <math>(1,2,3,4,5,6)</math> || 120 || <math>\! \frac{6!}{6}</math> || odd || 6
+! Total (11 rows = 11 conjugacy classes) || -- || -- || -- || 720 (equals order of the whole group) || -- || odd: 360, 5 classes<br>even;no: 216, 5 classes<br>even;yes;yes: 144, 1 class || order 1: 1, order 2: 75, order 3: 80, order 4: 180, order 5: 144, order 6: 240 || --
 |}
@@ Line 59: / Line 67: @@
 | 3 + 1 + 1 + 1, 3 + 3|| <math>(1,2,3)</math>, <math>(1,2,3)(4,5,6)</math> || 40 || 2 || 80 || 3
 |-
-| 4 + 1 + 1, 4 + 2 || <math>(1,2,3,4)</math>, <math>(1,2,3,4)(5,6)</math> || 90 || 2 || 180 || 4
+| 4 + 1 + 1 || <math>(1,2,3,4)</math> || 90 || 1 || 90 || 4
+|-
+| 4 + 2 || <math>(1,2,3,4)(5,6)</math> || 90 || 1 || 90 || 4
 |-
 | 5 + 1 || <math>(1,2,3,4,5)</math> || 144 || 1 || 144 || 5
@@ Line 66: / Line 76: @@
 |-
 | 2 + 2 + 1 + 1 || <math>(1,2)(3,4)</math> || 45 || 1 || 45 || 2
+|-
+! Total (8 classes) || -- || -- || 11 || 720 || --
 |}
 <section end="automorphism class structure"/>