Difference between revisions of "Symmetric group:S5"
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+ | {{perspectives}} | ||
{{particular group}} | {{particular group}} | ||
− | + | [[importance rank::2| ]] | |
==Definition== | ==Definition== | ||
− | + | The symmetric group <math>S_5</math> is defined in the following equivalent ways: | |
− | + | # It is the group of all permutations on a set of five elements, i.e., it is the [[member of family::symmetric group]][[member of family::symmetric group on finite set| ]] of degree five. In particular, it is a [[member of family::symmetric group of prime degree]] and [[member of family::symmetric group of prime power degree]]. With this interpretation, it is denoted <math>S_5</math> or <math>\operatorname{Sym}(5)</math>. | |
+ | # It is the [[member of family::projective general linear group]] of [[member of family::projective general linear group of degree two|degree two]] over [[field:F5|the field of five elements]], i.e., <math>PGL(2,5)</math>. | ||
+ | # It is the [[member of family::projective semilinear group]] of [[member of family::projective semilinear group of degree two|degree two]] over [[field:F4|the field of four elements]], i.e., <math>P\Gamma L(2,4)</math>. | ||
===Presentation=== | ===Presentation=== | ||
− | == | + | ===Equivalence of definitions=== |
− | |||
− | + | * [[PGammaL(2,4) is isomorphic to S5]] | |
+ | * [[PGL(2,5) is isomorphic to S5]] | ||
− | + | ==Arithmetic functions== | |
− | {{ | + | {{compare and contrast arithmetic functions|order = 120}} |
− | {{not | + | {| class="sortable" border="1" |
+ | ! Function !! Value !! Similar groups !! Explanation | ||
+ | |- | ||
+ | | {{arithmetic function value order|120}} || As <math>\! S_n, n = 5:</math> <math>\! n! = 5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120</math> <br><br> As <math>\! PGL(2,q), q = 5</math> (see [[order formulas for linear groups of degree two]]): <math>\! q^3 - q = q(q-1)(q+1) = 5^3 - 5 = 5(4)(6) = 120</math><br>As <math>\! P\Gamma L(2,q), q = p^r, q = 4, p = 2, r = 2</math> (see [[order formulas for linear groups of degree two]]): <math>\! r(q^3 - q) = 2(4^3 - 4) = 2(60) = 120</math>.<br><br>See [[element structure of symmetric group:S5#Order computation]] for more information. | ||
+ | |- | ||
+ | | {{arithmetic function value exponent given order|60|120}} || As <math>\! S_n, n = 5:</math> <math>\! \operatorname{lcm} \{ 1,2,\dots,n \} = \operatorname{lcm} \{1,2,3,4,5 \} = 60</math> <br><br> As <math>\! PGL(2,q), q = p = 5:</math> (where <math>\! p</math> is the underlying prime for <math>\! q</math>) <math>\! p(q^2 - 1)/2 = 5 \cdot (5^2 - 1)/2 = 60</math> | ||
+ | |- | ||
+ | | [[derived length]] || -- || || not a solvable group. | ||
+ | |- | ||
+ | | [[nilpotency class]] || -- || ||not a nilpotent group. | ||
+ | |- | ||
+ | | {{arithmetic function value given order|Frattini length|1|120}} || [[Frattini-free group]]: intersection of maximal subgroups is trivial. | ||
+ | |- | ||
+ | | {{arithmetic function value given order|minimum size of generating set|2|120}} || <math>\! (1,2), (1,2,3,4,5)</math>; see also [[symmetric group on a finite set is 2-generated]] | ||
+ | |- | ||
+ | | {{arithmetic function value given order|subgroup rank of a group|2|120}} || | ||
+ | |- | ||
+ | | {{arithmetic function value given order|max-length of a group|5|120}} || | ||
+ | |- | ||
+ | | {{arithmetic function value given order|number of subgroups|156|120}} || | ||
+ | |- | ||
+ | | {{arithmetic function value given order|number of conjugacy classes|7|120}} || As <math>\! S_k, k = 5</math>: the number of conjugacy classes is <math>\! p(k) = p(5) = 7</math>, where <math>p</math> is the [[number of unordered integer partitions]]. See also [[cycle type determines conjugacy class]] <br><br> As <math>\! PGL(2,q), q = 5:</math> <math>\! q + 2 = 5 + 2 = 7</math> | ||
+ | |- | ||
+ | | {{arithmetic function value given order|number of conjugacy classes of subgroups|19|120}} || | ||
+ | |} | ||
− | + | ==Group properties== | |
− | {{ | + | {{quotation|'''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]]''.}} |
− | <math> | + | ===Basic properties=== |
− | + | {| class="sortable" border="1" | |
+ | !Property !! Meaning !! Satisfied? !! Explanation !! Comment | ||
+ | |- | ||
+ | |[[Dissatisfies property::abelian group]] || any two elements commute || No || <math>(1,2)</math>, <math>(1,3)</math> don't commute || <math>S_n</math> is non-abelian, <math>n \ge 3</math>. | ||
+ | |- | ||
+ | |[[Dissatisfies property::nilpotent group]] || has a [[central series]] || No || [[Centerless group|Centerless]]: The [[center]] is trivial || <math>S_n</math> is non-nilpotent, <math>n \ge 3</math>. | ||
+ | |- | ||
+ | |[[Dissatisfies property::metacyclic group]] || has a [[cyclic normal subgroup]] with a cyclic [[quotient group]] || No || No [[cyclic normal subgroup]] || <math>S_n</math> is not metacyclic, <math>n \ge 4</math>. | ||
+ | |- | ||
+ | |[[Dissatisfies property::supersolvable group]] || has a [[normal series]] of finite length with all successive quotients cyclic groups. || No || No [[cyclic normal subgroup]] || <math>S_n</math> is not supersolvable, <math>n \ge 4</math>. | ||
+ | |- | ||
+ | |[[Dissatisfies property::solvable group]] || has a [[normal series]] of finite length with all successive quotients abelian groups. || No || The subgroup <math>A_5</math> [[A5 is simple|is simple non-abelian]] || <math>A_n</math> is simple and hence <math>S_n</math> not solvable, <math>n \ge 5</math>. | ||
+ | |- | ||
+ | |[[Dissatisfies property::simple group]] || has no proper nontrivial [[normal subgroup]]. || No || has a proper nontrivial normal subgroup [[A5 in S5]]. || | ||
+ | |- | ||
+ | |[[Satisfies property::almost simple group]] || contains a centralizer-free [[simple normal subgroup]]. || Yes || It contains a centralizer-free [[simple normal subgroup]], namely [[A5 in S5]]. || [[symmetric groups are almost simple]] for degree 5 or higher. | ||
+ | |- | ||
+ | |[[Dissatisfies property::perfect group]] || equals its own [[derived subgroup]] || No || Its derived subgroup is [[A5 in S5]] and abelianization is [[cyclic group:Z2]]. || | ||
+ | |- | ||
+ | |[[Dissatisfies property::quasisimple group]] || perfect, and [[inner automorphism group]] is simple non-abelian. || No || Follows from not being perfect. || | ||
+ | |- | ||
+ | |[[Satisfies property::centerless group]] || [[center]] is trivial. || Yes || [[symmetric groups are centerless]] || [[symmetric groups are centerless]] for degree other than two. | ||
+ | |- | ||
+ | |[[Satisfies property::complete group]] || centerless, and every automorphism is inner. || Yes || Centerless and every automorphism's inner || [[Symmetric groups are complete]] except the ones of degree <math>2,6</math>. | ||
+ | |} | ||
− | { | + | ===Other properties=== |
+ | {| class="sortable" border="1" | ||
+ | !Property !! Satisfied? !! Explanation !! Comment | ||
+ | |- | ||
+ | |[[Satisfies property::T-group]] || Yes || || | ||
+ | |- | ||
+ | |[[Satisfies property::HN-group]] || Yes || || | ||
− | The | + | |- |
+ | |[[Satisfies property::monolithic group]] || Yes || Monolith is the alternating group || All symmetric groups are monolithic; <math>n=4</math> is the only case the monolith is not the alternating group. | ||
+ | |- | ||
+ | |[[Satisfies property::one-headed group]] || Yes || The alternating group is the unique [[maximal normal subgroup]] || True for all <math>n > 1</math>. | ||
+ | |- | ||
+ | |[[satisfies property::N-group]] || Yes || See [[classification of symmetric groups that are N-groups]] || True only for <math>n \le 6</math>. | ||
+ | |} | ||
− | == | + | ==Elements== |
− | + | {{further|[[element structure of symmetric group:S5]]}} | |
− | + | ===Upto conjugacy=== | |
− | + | For convenience, we take the underlying set to be <math>\{ 1,2,3,4,5 \}</math>. | |
− | <math> | + | There are seven conjugacy classes, corresponding to the [[set of unordered integer partitions|unordered integer partitions]] of <math>5</math> (for more information, refer [[cycle type determines conjugacy class]]). We use the notation of the [[cycle decomposition for permutations]]: |
− | + | {{#lst:element structure of symmetric group:S5|conjugacy class structure}} | |
− | |||
− | |||
+ | ===Upto automorphism=== | ||
− | + | <math>S_5</math> 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, <math>S_n</math> is complete for <math>n \ne 2, 6</math>. See [[symmetric groups on finite sets are complete]]. | |
− | + | ==Endomorphisms== | |
− | + | {{further|[[endomorphism structure of symmetric group:S5]]}} | |
− | + | ===Summary information=== | |
− | + | {{#lst:endomorphism structure of symmetric group:S5|summary}} | |
− | + | ===Automorphisms=== | |
− | + | Since <math>S_5</math> is a complete group, it is isomorphic to its automorphism group, where each element of <math>S_5</math> acts on <math>S_5</math> by conjugation. {{further|[[symmetric groups on finite sets are complete]]}} | |
− | + | ==Subgroups== | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | {{further|[[Subgroup structure of symmetric group:S5]]}} | |
+ | {{#lst:subgroup structure of symmetric group:S5|summary}} | ||
− | + | ==Linear representation theory== | |
+ | {{further|[[linear representation theory of symmetric group:S5]]}} | ||
− | + | ===Summary=== | |
− | + | {{#lst:linear representation theory of symmetric group:S5|summary}} | |
− | + | ===Character table=== | |
− | + | {{#lst:linear representation theory of symmetric group:S5|character table}} | |
− | + | ==GAP implementation== | |
− | + | {{GAP ID|120|34}} | |
− | + | ===Other descriptions=== | |
− | { | + | {| class="sortable" border="1" |
+ | ! Description !! Functions used | ||
+ | |- | ||
+ | | <tt>SymmetricGroup(5)</tt> || [[GAP:SymmetricGroup|SymmetricGroup]] | ||
+ | |- | ||
+ | | <tt>PGL(2,5)</tt> || [[GAP:PGL|PGL]] | ||
+ | |} |
Latest revision as of 16:24, 23 July 2013
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Contents
Definition
The symmetric group is defined in the following equivalent ways:
- 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 or .
- It is the projective general linear group of degree two over the field of five elements, i.e., .
- It is the projective semilinear group of degree two over the field of four elements, i.e., .
Presentation
Equivalence of definitions
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
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 | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
abelian group | any two elements commute | No | , don't commute | is non-abelian, . |
nilpotent group | has a central series | No | Centerless: The center is trivial | is non-nilpotent, . |
metacyclic group | has a cyclic normal subgroup with a cyclic quotient group | No | No cyclic normal subgroup | is not metacyclic, . |
supersolvable group | has a normal series of finite length with all successive quotients cyclic groups. | No | No cyclic normal subgroup | is not supersolvable, . |
solvable group | has a normal series of finite length with all successive quotients abelian groups. | No | The subgroup is simple non-abelian | is simple and hence not solvable, . |
simple group | has no proper nontrivial normal subgroup. | No | has a proper nontrivial normal subgroup A5 in S5. | |
almost simple group | contains a centralizer-free simple normal subgroup. | 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 | equals its own derived subgroup | No | Its derived subgroup is A5 in S5 and abelianization is cyclic group:Z2. | |
quasisimple group | perfect, and inner automorphism group is simple non-abelian. | No | Follows from not being perfect. | |
centerless group | center is trivial. | Yes | symmetric groups are centerless | symmetric groups are centerless for degree other than two. |
complete group | centerless, and every automorphism is inner. | Yes | Centerless and every automorphism's inner | Symmetric groups are complete except the ones of degree . |
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; 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-group | Yes | See classification of symmetric groups that are N-groups | True only for . |
Elements
Further information: element structure of symmetric group:S5
Upto conjugacy
For convenience, we take the underlying set to be .
There are seven conjugacy classes, corresponding to the unordered integer partitions of (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 | even; no | 1 | ||
2 + 1 + 1 + 1 | 2 (1 time), 1 (3 times) | transposition: one 2-cycle, three fixed point | 10 | or , also in this case | odd | 2 | ||
3 + 1 + 1 | 3 (1 time), 1 (2 times) | one 3-cycle, two fixed points | 20 | or | even; no | 3 | ||
2 + 2 + 1 | 2 (2 times), 1 (1 time) | double transposition: two 2-cycles, one fixed point | 15 | or | even; no | 2 | ||
4 + 1 | 4 (1 time), 1 (1 time) | one 4-cycle, one fixed point | 30 | or | odd | 4 | ||
3 + 2 | 3 (1 time), 2 (1 time) | one 3-cycle, one 2-cycle | 20 | or | odd | 6 | ||
5 | 5 (1 time) | one 5-cycle | 24 | or | even; yes; yes | 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) |
[SHOW MORE] |
Upto automorphism
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, is complete for . 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 is a complete group, it is isomorphic to its automorphism group, where each element of acts on 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 : 1,2,6,30,156,1455,11300, 151221 |
Number of conjugacy classes of subgroups | 19 Compared with , : 1,2,4,11,19,56,96,296,554,1593 |
Number of automorphism classes of subgroups | 19 Compared with , : 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 | -Hall subgroup: S4 in S5 (order 24) No -Hall subgroup or -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.
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 or ) | 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) | -- ring of integers |
Smallest field of realization for all irreducible representations, i.e., smallest splitting field (characteristic zero) | -- 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) | (size 10) | (size 15) | (size 20) | (size 20) | (size 24) | (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 :
gap> G := SmallGroup(120,34);
Conversely, to check whether a given group 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 |