Difference between revisions of "Symmetric group:S5"
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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]]: | 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]]: | ||
− | + | {| class="sortable" border="1" | |
− | + | ! Partition !! Verbal description of cycle type !! Representative element with the cycle type !! Size of conjugacy class !! Formula calculating size !! Even or odd? | |
− | + | |- | |
− | + | | 1 + 1 + 1 + 1 + 1 || five fixed points || <math>()</math> -- the identity element || 1 || <math>\! \frac{5!}{(1)^5(5!)</math> || even | |
− | + | |- | |
− | + | | 2 + 1 + 1 + 1 || transposition: one 2-cycle, three fixed point || <math>(1,2)</math> || 10 || <math>\! \frac{5!}{(2)(1)^3(3!)}</math>, also <math>\binom{5}{2}</math> in this case || odd | |
− | + | |- | |
− | + | | 3 + 1 + 1 || one 3-cycle, two fixed points ||<math>(1,2,3)</math> || 20 || <math>\! \frac{5!}{(3)(1)^2(2!)}</math> || even | |
− | + | |- | |
+ | | 2 + 2 + 1 || double transposition: two 2-cycles, one fixed point || <math>(1,2)(3,4)</math> || 15 || <math>\! \frac{5!}{(2)^2(2!)(1)}</math> || even | ||
+ | |- | ||
+ | | 4 + 1 || one 4-cycle, one fixed point || <math>(1,2,3,4)</math> || 30 || <math>\! \frac{5!}{(4)(1)}</math> || odd | ||
+ | |- | ||
+ | | 3 + 2 || one 3-cycle, one 2-cycle || <math>(1,2,3)(4,5)</math> || 20 || <math>\! \frac{5!}{(3)(2)}</math> || odd | ||
+ | |- | ||
+ | | 5 || one 5-cycle || <math>(1,2,3,4,5)</math> || 24 || <math>\! \frac{5!}{5}</math> || even | ||
+ | |} | ||
===Upto automorphism=== | ===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. | <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. | ||
+ | |||
==Endomorphisms== | ==Endomorphisms== | ||
Revision as of 23:20, 30 March 2010
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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.
- It is the projective general linear group of degree two over the field of five elements, i.e., .
Presentation
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order | 120 | . |
exponent | 60 | Elements of order . |
derived length | -- | not a solvable group. |
nilpotency class | -- | not a nilpotent group. |
Frattini length | 1 | Frattini-free group: intersection of maximal subgroups is trivial. |
minimum size of generating set | 2 | |
subgroup rank | 2 | |
max-length | 5 | -- |
number of subgroups | 156 | -- |
number of conjugacy classes | 7 | |
number of conjugacy classes of subgroups | 19 |
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.
Property | Satisfied | Explanation | Comment |
---|---|---|---|
Abelian group | No | , don't commute | is non-abelian, . |
Nilpotent group | No | Centerless: The center is trivial | is non-nilpotent, . |
Metacyclic group | No | No cyclic normal subgroup | is not metacyclic, . |
Supersolvable group | No | No cyclic normal subgroup | is not supersolvable, . |
Solvable group | No | The subgroup is simple non-abelian | is simple and hence not solvable, . |
T-group | Yes | ||
HN-group | Yes | ||
Complete group | Yes | Centerless and every automorphism's inner | Symmetric groups are complete except the ones of degree . |
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 . |
Elements
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 | Verbal description of cycle type | Representative element with the cycle type | Size of conjugacy class | Formula calculating size | Even or odd? |
---|---|---|---|---|---|
1 + 1 + 1 + 1 + 1 | five fixed points | -- the identity element | 1 | Failed to parse (syntax error): \! \frac{5!}{(1)^5(5!) | even |
2 + 1 + 1 + 1 | transposition: one 2-cycle, three fixed point | 10 | , also in this case | odd | |
3 + 1 + 1 | one 3-cycle, two fixed points | 20 | even | ||
2 + 2 + 1 | double transposition: two 2-cycles, one fixed point | 15 | even | ||
4 + 1 | one 4-cycle, one fixed point | 30 | odd | ||
3 + 2 | one 3-cycle, one 2-cycle | 20 | odd | ||
5 | one 5-cycle | 24 | even |
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.
Endomorphisms
Automorphisms
Since is a complete group, it is isomorphic to its automorphism group, where each element of acts on by conjugation.
Endomorphisms
admits three kinds of endomorphisms (that is, it admits more endomorphisms, but any endomorphism is equivalent via an automorphism to one of these three):
- The endomorphism to the trivial group
- The identity map
- The endomorphism to a group of order two, given by the sign homomorphism
Subgroups
Further information: Subgroup structure of symmetric group:S5
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
The group can also be defined using GAP's SymmetricGroup function as:
SymmetricGroup(5)