Symmetric group:S6: Difference between revisions

Latest revision as of 21:23, 15 December 2023

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The symmetric group $S_{6}$ , called the symmetric group of degree six, is defined in the following equivalent ways:

It is the symmetric group on a set of size six. In particular, it is a symmetric group on finite set.
It is the symplectic group $S p (4, 2)$ , and hence also the projective symplectic group $P S p (4, 2)$ (see isomorphism between symplectic and projective symplectic group in characteristic two).

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 720#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set)	720	groups with same order	As $S_{n}, n = 6 :$ $n! = 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720$ As $S p (2 m, q)$ , $m = 2, q = 2$ (see order formulas for symplectic groups): $q^{m^{2}} \prod_{i = 1}^{m} (q^{2 i} - 1)$ which becomes $2^{4} (2^{2} - 1) (2^{4} - 1) = 16 \cdot 3 \cdot 15 = 720$
exponent	60	groups with same order and exponent \| groups with same exponent	As $S_{n}, n = 6 :$ $l c m {1, 2, 3, 4, 5, 6} = 60$
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; see also symmetric groups are Frattini-free
minimum size of generating set	2	groups with same order and minimum size of generating set \| groups with same minimum size of generating set	$(1, 2), (1, 2, 3, 4, 5, 6)$ ; see also symmetric group on a finite set is 2-generated
subgroup rank of a group	3	groups with same order and subgroup rank of a group \| groups with same subgroup rank of a group	The group elementary abelian group:E8 can be embedded in this group as $⟨ (1, 2), (3, 4), (5, 6) ⟩$
max-length of a group	6	groups with same order and max-length of a group \| groups with same max-length of a group	This is a rare example of a small group whose max-length is less than the sum of the exponents of all prime divisors
composition length	2	groups with same order and composition length \| groups with same composition length	The subgroup alternating group:A6 is simple and normal (see alternating groups are simple) and the quotient is simple (cyclic of order two)
chief length	2	groups with same order and chief length \| groups with same chief length	The subgroup alternating group:A6 is simple and normal (see alternating groups are simple) and the quotient is simple (cyclic of order two)

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation for function value
number of conjugacy classes	11	groups with same order and number of conjugacy classes \| groups with same number of conjugacy classes	As $S_{n}, n = 6 :$ the number of conjugacy classes is $p (n) = p (6) = 11$ , where $p$ is the number of unordered integer partitions; see cycle type determines conjugacy class. As $S p (4, q), q = 2$ (even): $q^{2} + 2 q + 3 = 2^{2} + 2 (2) + 3 = 11$ . More here See element structure of symmetric group:S6
number of conjugacy classes of subgroups	56	groups with same order and number of conjugacy classes of subgroups \| groups with same number of conjugacy classes of subgroups
number of subgroups	1455	groups with same order and number of subgroups \| groups with same number of subgroups

Group properties

Property	Satisfied?	Explanation
abelian group	No	$(1, 2)$ and $(2, 3)$ don't commute. In fact, $S_{n}$ is non-abelian for $n \geq 3$ .
nilpotent group	No
solvable group	No
simple group	No	has alternating group:A6 as a proper nontrivial normal subgroup
almost simple group	Yes	sandwiched between the simple group alternating group:A6 and the automorphism group thereof. See also symmetric groups are almost simple.
quasisimple group	No
one-headed group	Yes	alternating group:A6 is the unique maximal normal subgroup
monolithic group	Yes	alternating group:A6 is the unique minimal normal subgroup
T-group	Yes
rational representation group	Yes	See symmetric groups are rational representation, linear representation theory of symmetric groups
rational group	Yes	See symmetric groups are rational
N-group	Yes	See classification of symmetric groups that are N-groups

Elements

Further information: element structure of symmetric group:S6

Up to conjugacy

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

There are eleven conjugacy classes, corresponding to the unordered integer partitions of $6$ (for more information, refer cycle type determines conjugacy class).

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	--

Up to automorphism

The outer automorphism group has order two, and it swaps some conjugacy classes. Below are the equivalence classes up to automorphisms.

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	--

Subgroups

Further information: subgroup structure of symmetric group:S6

Quick summary

Item	Value
Number of subgroups	1455 Compared with $S_{n}, n = 1, 2, 3, \dots$ : 1, 2, 6, 30, 156, 1455, 11300, 151221, ...
Number of conjugacy classes of subgroups	56 Compared with $S_{n}, n = 1, 2, 3, \dots$ : 1, 2, 4, 11, 19, 56, 96, 296, ...
Number of automorphism classes of subgroups	37 Compared with $S_{n}, n = 1, 2, 3, \dots$ : 1, 2, 4, 11, 19, 37, 96, 296, ...
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems	2-Sylow: direct product of D8 and Z2 (order 16), Sylow number is 45 3-Sylow: elementary abelian group:E9 (order 9), Sylow number is 10 5-Sylow: cyclic group:Z5 (order 5), Sylow number is 36
Hall subgroups	No Hall subgroups other than the Sylow subgroups, whole group, and trivial subgroup. In particular, there is no ${2, 3}$ -Hall subgroup, ${2, 5}$ -Hall subgroup, and ${3, 5}$ -Hall subgroup.
maximal subgroups	maximal subgroups have order 48, 72, 120, and 360
normal subgroups	The only normal subgroups are the whole group, the trivial subgroup, and alternating group:A6 as A6 in S6.

Linear representation theory

Further information: linear representation theory of symmetric group:S6

Summary

Item	Summary
Degrees of irreducible representations over a splitting field	1,1,5,5,5,5,9,9,10,10,16 maximum: 16, lcm: 720, number: 11, sum of squares: 720
Schur index values of irreducible representations	1,1,1,1,1,1,1,1,1,1,1
Smallest ring of realization for all irreducible representations (characteristic zero)	$Z$ -- ring of integers
Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero)	$Q$ -- hence it is a rational representation group
Criterion for a field to be a splitting field	Any field of characteristic not 2,3, or 5
Smallest size splitting field	field:F7

Character table

Representation/conjugacy class representative and size	$()$ (size 1)	$(1, 2)$ (size 15)	$(1, 2) (3, 4) (5, 6)$ (size 15)	$(1, 2, 3)$ (size 40)	$(1, 2, 3) (4, 5, 6)$ (size 40)	$(1, 2) (3, 4)$ (size 45)	$(1, 2, 3, 4)$ (size 90)	$(1, 2, 3, 4) (5, 6)$ (size 90)	$(1, 2, 3) (4, 5)$ (size 120)	$(1, 2, 3, 4, 5, 6)$ (size 120)	$(1, 2, 3, 4, 5)$ -- size 144
trivial	1	1	1	1	1	1	1	1	1	1	1
sign	1	-1	-1	1	1	1	-1	1	-1	-1	1
standard	5	3	-1	2	-1	1	1	-1	0	-1	0
product of standard and sign	5	-3	1	2	-1	1	-1	-1	0	1	0
other five-dimensional irreducible	5	-1	3	-1	2	1	1	-1	-1	0	0
other five-dimensional irreducible	5	1	-3	-1	2	1	-1	-1	1	0	0
nine-dimensional irreductible	9	3	3	0	0	1	-1	1	0	0	-1
product of nine-dimensional irreductible and sign	9	-3	-3	0	0	1	1	1	0	0	-1
second exterior power of standard	10	2	-2	1	1	-2	0	0	-1	1	0
third exterior power of standard	10	-2	2	1	1	-2	0	0	1	-1	0
sixteen-dimensional irreductible	16	0	0	-2	-2	0	0	0	0	0	1

GAP implementation

Group ID

This finite group has order 720 and has ID 763 among the groups of order 720 in GAP's SmallGroup library. For context, there are groups of order 720. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(720,763)

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

gap> G := SmallGroup(720,763);

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

IdGroup(G) = [720,763]

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(6)`	SymmetricGroup
`Sp(4,2)`	Sp
`PSp(4,2)`	PSp

@@ Line 1: / Line 1: @@
 {{particular group}}
+[[Category:Symmetric groups]]
+[[importance rank::2| ]]
 ==Definition==
 The symmetric group <math>S_6</math>, called the '''symmetric group of degree six''', is defined in the following equivalent ways:
-* It is the [[member of family::symmetric group]] on a set of size six. Other equivalent definitions include:
+* It is the [[member of family::symmetric group]] on a set of size six. In particular, it is a [[member of family::symmetric group on finite set]].
-* It is the [[member of family::projective general linear group]] <math>PGL(2,9)</math>.
+* It is the [[member of family::symplectic group]] <math>Sp(4,2)</math>, and hence also the [[member of family::projective symplectic group]] <math>PSp(4,2)</math> (see [[isomorphism between symplectic and projective symplectic group in characteristic two]]).
-* It is the [[member of family::symplectic group]] <math>Sp(4,2)</math>, and hence also the [[member of family::projective symplectic group]] <math>PSp(4,2)</math>.
+==Arithmetic functions==
+{{compare and contrast arithmetic functions|order = 720}}
+===Basic arithmetic functions===
+{| class="sortable" border="1"
+! Function !! Value !! Similar groups !! Explanation for function value
+|-
+| {{arithmetic function value order|720}} || As <math>\!S_n, n = 6:</math> <math>n! = 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720</math><br>As <math>\! Sp(2m,q)</math>, <math>\! m = 2, q = 2</math> (see [[order formulas for symplectic groups]]): <math>q^{m^2}\prod_{i=1}^m (q^{2i} - 1)</math> which becomes <math>2^4(2^2 - 1)(2^4 - 1) = 16 \cdot 3 \cdot 15 = 720</math>
+|-
+| {{arithmetic function value exponent given order|60|720}} || As <math>\! S_n, n = 6:</math> <math>\operatorname{lcm} \{ 1,2,3,4,5,6 \} = 60</math>
+|-
+| [[derived length]] || -- || || not a [[solvable group]]
+|-
+| [[nilpotency class]] || -- || || not a [[nilpotent group]]
+|-
+| {{arithmetic function value given order|Frattini length|1|720}} || [[Frattini-free group]]; see also [[symmetric groups are Frattini-free]]
+|-
+| {{arithmetic function value given order|minimum size of generating set|2|720}} || <math>\! (1,2), (1,2,3,4,5,6)</math>; see also [[symmetric group on a finite set is 2-generated]]
+|-
+| {{arithmetic function value given order|subgroup rank of a group|3|720}} || The group [[elementary abelian group:E8]] can be embedded in this group as <math>\langle (1,2), (3,4), (5,6) \rangle</math>
+|-
+| {{arithmetic function value given order|max-length of a group|6|720}} || This is a rare example of a small group whose max-length is less than the sum of the exponents of all prime divisors
+|-
+| {{arithmetic function value given order|composition length|2|720}} || The subgroup [[alternating group:A6]] is simple and normal (see [[alternating groups are simple]]) and the quotient is simple ([[cyclic group:Z2|cyclic of order two]])
+|-
+| {{arithmetic function value given order|chief length|2|720}} || The subgroup [[alternating group:A6]] is simple and normal (see [[alternating groups are simple]]) and the quotient is simple ([[cyclic group:Z2|cyclic of order two]])
+|}
+===Arithmetic functions of a counting nature===
+{| class="sortable" border="1"
+! Function !! Value !! Similar groups !! Explanation for function value
+|-
+| {{arithmetic function value given order|number of conjugacy classes|11|720}} || As <math>\! S_n, n = 6:</math> the number of conjugacy classes is <math>p(n) = p(6) = 11</math>, where <math>p</math> is the [[number of unordered integer partitions]]; see [[cycle type determines conjugacy class]].<br>As <math>Sp(4,q), q = 2</math> (even): <math>q^2 + 2q + 3 = 2^2 + 2(2) + 3 = 11</math>. More [[element structure of symplectic group of degree four over a finite field|here]]<br>See [[element structure of symmetric group:S6]]
+|-
+| {{arithmetic function value given order|number of conjugacy classes of subgroups|56|720}} ||
+|-
+| {{arithmetic function value given order|number of subgroups|1455|720}} ||
+|}
+==Group properties==
+{| class="sortable" border="1"
+! Property !! Satisfied? !! Explanation
+|-
+| [[dissatisfies property::abelian group]] || No || <math>(1,2)</math> and <math>(2,3)</math> don't commute. In fact, <math>S_n</math> is non-abelian for <math>n \ge 3</math>.
+|-
+| [[dissatisfies property::nilpotent group]] || No ||
+|-
+| [[dissatisfies property::solvable group]] || No ||
+|-
+| [[dissatisfies property::simple group]] || No || has [[alternating group:A6]] as a proper nontrivial normal subgroup
+|-
+| [[satisfies property::almost simple group]] || Yes || sandwiched between the simple group [[alternating group:A6]] and the automorphism group thereof. See also [[symmetric groups are almost simple]].
+|-
+| [[dissatisfies property::quasisimple group]] || No ||
+|-
+| [[satisfies property::one-headed group]] || Yes || [[alternating group:A6]] is the unique [[maximal normal subgroup]]
+|-
+| [[satisfies property::monolithic group]] || Yes || [[alternating group:A6]] is the unique [[minimal normal subgroup]]
+|-
+| [[satisfies property::T-group]] || Yes ||
+|-
+| [[satisfies property::rational representation group]] || Yes || See [[symmetric groups are rational representation]], [[linear representation theory of symmetric groups]]
+|-
+| [[satisfies property::rational group]] || Yes || See [[symmetric groups are rational]]
+|-
+| [[satisfies property::N-group]] || Yes || See [[classification of symmetric groups that are N-groups]]
+|}
 ==Elements==
-===Upto conjugacy===
+{{further|[[element structure of symmetric group:S6]]}}
+===Up to conjugacy===
 For convenience, we take the underlying set here as <math>\{ 1,2,3,4,5,6 \}</math>.
-There are eleven conjugacy classes, corresponding to the unordered integer partitions of <math>6</math> (for more information, refer [[cycle type determines conjugacy class]]):
+There are eleven conjugacy classes, corresponding to the unordered integer partitions of <math>6</math> (for more information, refer [[cycle type determines conjugacy class]]).
+{{#lst:element structure of symmetric group:S6|conjugacy class structure}}
+===Up to automorphism===
+The [[outer automorphism group]] has order two, and it swaps some conjugacy classes. Below are the equivalence classes up to automorphisms.
+{{#lst:element structure of symmetric group:S6|automorphism class structure}}
+==Subgroups==
-# <math>6 = 1 + 1 + 1 + 1 + 1 + 1</math>, i.e., six cycles of size one: The identity element. (1)
+{{further|[[subgroup structure of symmetric group:S6]]}}
-# <math>6 = 2 + 1 + 1 + 1 + 1</math>, i.e., one <math>2</math>-cycle and four fixed points: The transpositions, such as <math>(1,2)</math>. (15)
-# <math>6 = 3 + 1 + 1 + 1</math>, i.e., one <math>3</math>-cycle and three fixed points: The <math>3</math>-cycles, such as <math>(1,2,3)</math>. (40)
-# <math>6 = 4 + 1 + 1</math>: The <math>4</math>-cycles, such as <math>(1,2,3,4)</math>. (90)
-# <math>6 = 5 + 1</math>: The <math>5</math>-cycles, such as <math>(1,2,3,4,5)</math>. (144)
-# <math>6 = 6</math>: The <math>6</math>-cycles, such as <math>(1,2,3,4,5,6)</math>. (120)
-# <math>6= 2 + 2 + 1 + 1</math>, i.e., two <math>2</math>-cycles, two fixed points: The double transpositions, such as <math>(1,2)(3,4)</math>. (45)
-# <math>6 = 2 + 2 + 2</math>, i.e., three <math>2</math>-cycles: The triple transpositions, such as <math>(1,2)(3,4)(5,6)</math>. (15)
-# <math>6 = 3 + 2 + 1</math>, i.e., one <math>3</math>-cycle, one <math>2</math>-cycle: Permutations such as <math>(1,2,3)(4,5)</math>.(120)
-# <math>6 = 3 + 3</math>, i.e., two <math>3</math>-cycles: Permutations such as <math>(1,2,3)(4,5,6)</math>. (40)
-# <math>6 = 4 + 2</math>: Permutations such as <math>(1,2,3,4)(5,6)</math>. (90)
-Of these, types (1), (3), (5), (7), (10), (11) are conjugacy classes of [[even permutation]]s -- together these form the [[alternating group:A6|alternating group of degree six]]. The remaining types: (2), (4), (6), (8), (9), are [[odd permutation]]s.
+{{#lst:subgroup structure of symmetric group:S6|summary}}
-===Upto automorphism===
+==Linear representation theory==
-Under automorphisms, the following types get merged:
+{{further|[[linear representation theory of symmetric group:S6]]}}
-* Types (2) and (8): The transposition <math>(1,2)</math> is related by an outer automorphism to the triple transposition <math>(1,2)(3,4)(5,6)</math>.
+===Summary===
-* Types (3) and (10): The <math>3</math>-cycle <math>(1,2,3)</math> is related by an outer automorphism to the permutation <math>(1,2,3)(4,5,6)</math>.
+{{#lst:linear representation theory of symmetric group:S6|summary}}
-* Types (4) and (11): The <math>4</math>-cycle <math>(1,2,3,4)</math> is related by an outer automorphism to the pemrutation <math>(1,2,3,4)(5,6)</math>.
-* Types (6) and (9): The <math>6</math>-cycle <math>(1,2,3,4,5,6)</math> is related by an outer automorphism to the permutation <math>(1,2,3)(4,5)</math>.
-The types (1), (5), and (7) remain unaffected: these conjugacy classes are also automorphism classes.
+===Character table===
+{{#lst:linear representation theory of symmetric group:S6|character table}}
 ==GAP implementation==
 {{GAP ID|720|763}}
+===Other descriptions===
+{| class="sortable" border="1"
+! Description !! Functions used
+|-
+| <tt>SymmetricGroup(6)</tt> || [[GAP:SymmetricGroup|SymmetricGroup]]
+|-
+| <tt>Sp(4,2)</tt> || [[GAP:Sp|Sp]]
+|-
+| <tt>PSp(4,2)</tt> || [[GAP:PSp|PSp]]
+|}