Element structure of symmetric group:S4
ALSO CHECK OUT: Quiz (multiple choice questions to test your understanding) |
This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S4.
View element structure of particular groups | View other specific information about symmetric group:S4
This article discusses the element structure of symmetric group:S4, the symmetric group of degree four. We denote its elements as acting on the set , written using cycle decompositions, with composition by function composition where functions act on the left.
Since this group is a complete group (i.e., every automorphism is inner and the center is trivial), the classification of elements up to conjugacy is the same as the classification up to automorphisms. Further, since cycle type determines conjugacy class for symmetric groups, the conjugacy classes are parametrized by cycle types, which in turn are parametrized by unordered integer partitions of .
This page concentrates on the more group-theoretic aspects of the element structure. For the more combinatorial aspects, see combinatorics of symmetric group:S4.
Summary
Item | Value |
---|---|
order of the whole group (total number of elements) | 24 prime factorization ![]() See order computation for more For other groups of the same order, see groups of order 24 |
conjugacy class sizes | 1,3,6,6,8 maximum: 8, number: 5, sum (equals order of group): 24, lcm: 24 See conjugacy class structure for more. |
number of conjugacy classes | 5 See element structure of symmetric group:S4#Number of conjugacy classes |
order statistics | 1 of order 1, 9 of order 2, 8 of order 3, 6 of order 4 maximum: 4, lcm (exponent of the whole group): 12 |
Family contexts
Note that if you go to the #Conjugacy class structure section of this article, you'll find a discussion of the conjugacy class structure with each of the below family interpretations.
Family name | Parameter values | General discussion of element structure of family |
---|---|---|
symmetric group | degree ![]() |
element structure of symmetric groups |
projective general linear group of degree two over a finite field | field:F3, i.e., the group is ![]() |
element structure of projective general linear group of degree two over a finite field |
general affine group of degree two over a finite field | field:F2, i.e., the group is ![]() |
element structure of general affine group of degree two over a finite field |
COMPARE AND CONTRAST: View element structure of groups of order 24 to compare and contrast the element structure with other groups of order 24.
Elements
Multiple ways of describing permutations
Note that the matrix for the right action is obtained by taking the transpose of the matrix for the left action. For the identity element and the elements of order 2, both matrices coincide.
Cycle decomposition notation | One-line notation, i.e., image of string ![]() |
Matrix (left action) | Matrix (right action) | Order of element (= lcm of cycle sizes) |
---|---|---|---|---|
![]() |
1234 | ![]() |
![]() |
1 |
![]() |
1243 | ![]() |
![]() |
2 |
![]() |
1324 | ![]() |
![]() |
2 |
![]() |
1342 | ![]() |
![]() |
3 |
![]() |
1423 | ![]() |
![]() |
3 |
![]() |
1432 | ![]() |
![]() |
2 |
![]() |
2134 | ![]() |
![]() |
2 |
![]() |
2143 | ![]() |
![]() |
2 |
![]() |
2314 | ![]() |
![]() |
3 |
![]() |
2341 | ![]() |
![]() |
4 |
![]() |
2413 | ![]() |
![]() |
4 |
![]() |
2431 | ![]() |
![]() |
3 |
![]() |
3124 | ![]() |
![]() |
3 |
![]() |
3142 | ![]() |
![]() |
4 |
![]() |
3214 | ![]() |
![]() |
2 |
![]() |
3241 | ![]() |
![]() |
3 |
![]() |
3412 | ![]() |
![]() |
2 |
![]() |
3421 | ![]() |
![]() |
4 |
![]() |
4123 | ![]() |
![]() |
4 |
![]() |
4132 | ![]() |
![]() |
3 |
![]() |
4213 | ![]() |
![]() |
3 |
![]() |
4231 | ![]() |
![]() |
2 |
![]() |
4312 | ![]() |
![]() |
4 |
![]() |
4321 | ![]() |
![]() |
2 |
Order computation
The symmetric group of degree four has order 24, with prime factorization . Below are listed various methods that can be used to compute the order, all of which should give the answer 24:
Family | Parameter values | Formula for order of a group in the family | Proof or justification of formula | Evaluation at parameter values (should give the answer 24) | Full interpretation of conjugacy class structure |
---|---|---|---|---|---|
symmetric group ![]() ![]() |
degree ![]() |
![]() |
See symmetric group, element structure of symmetric groups | ![]() |
#Interpretation as symmetric group |
projective general linear group of degree two over a finite field of size ![]() |
size ![]() ![]() |
![]() In factored form: ![]() |
See order formulas for linear groups of degree two, order formulas for linear groups, projective general linear group of degree two | ![]() ![]() In factored form: ![]() ![]() |
#Interpretation as projective general linear group of degree two |
general affine group of degree two over a finite field of size ![]() |
size ![]() ![]() ![]() |
![]() |
See order formulas for linear groups of degree two | ![]() |
#Interpretation as general affine group of degree two |
general semiaffine group of degree one over a finite field of size ![]() ![]() |
![]() ![]() ![]() |
![]() |
See order formulas for linear groups of degree one, general semiaffine group of degree one | ![]() |
#Interpretation as general semiaffine group of degree one |
projective special linear group of degree two over a finite discrete valuation ring with length ![]() ![]() |
![]() ![]() ![]() |
![]() ![]() |
See order formulas for linear groups of degree two | Here ![]() ![]() ![]() |
#Interpretation as projective special linear group of degree two |
von Dyck group with parameters ![]() |
![]() |
![]() |
See element structure of von Dyck groups | ![]() |
#Interpretation as von Dyck group |
triangle group with parameters ![]() |
![]() |
![]() |
See element structure of triangle groups | ![]() |
#Interpretation as triangle group |
Computation of prime powers in order
The prime factorization of the order is:
Family | Parameter values | Formula for order of a group in the family | Formula for the largest power of a generic prime ![]() |
Case ![]() |
Case ![]() |
Case ![]() |
---|---|---|---|---|---|---|
symmetric group ![]() ![]() |
degree ![]() |
![]() |
![]() |
![]() |
![]() |
no summands, so sum is 0 |
projective general linear group of degree two over a finite field of size ![]() |
size ![]() ![]() |
![]() In factored form: ![]() |
If ![]() ![]() ![]() If ![]() ![]() ![]() If ![]() ![]() ![]() Otherwise, if ![]() ![]() ![]() |
We use the ![]() ![]() |
We use the case that ![]() ![]() ![]() |
Since 3 - 1 = 2 and 3 + 1 = 4 are powers of 2, there are no primes involved other than 2 and 3. |
general affine group of degree two over a finite field of size ![]() |
size ![]() ![]() ![]() |
![]() |
If ![]() ![]() ![]() If ![]() ![]() ![]() If ![]() ![]() ![]() Otherwise, if ![]() ![]() ![]() ![]() Otherwise, if ![]() ![]() ![]() ![]() Otherwise, 0. |
![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
No primes other than 2 and 3 divide 2, 2 + 1, or 2 - 1. |
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
The conjugacy class sizes are .
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 any symmetric group, cycle type determines conjugacy class, i.e., the cycle type of a permutation (which describes the sizes of the cycles in a cycle decomposition of that permutation), determines its conjugacy class. In other words, two permutations are conjugate if and only if they have the same number of cycles of each size.
The cycle types (and hence the conjugacy classes) are parametrized by partitions of the size of the set. We describe the situation for this group:
Partition | Partition in grouped form | Verbal description of cycle type | Elements with the cycle type | Size of conjugacy class | Formula for size | Even or odd? If even, splits? If splits, real in alternating group? | Element order | Formula calculating element order |
---|---|---|---|---|---|---|---|---|
1 + 1 + 1 + 1 | 1 (4 times) | four cycles of size one each, i.e., four fixed points | ![]() |
1 | ![]() |
even; no | 1 | ![]() |
2 + 1 + 1 | 2 (1 time), 1 (2 times) | one transposition (cycle of size two), two fixed points | ![]() ![]() ![]() ![]() ![]() ![]() |
6 | ![]() ![]() |
odd | 2 | ![]() |
2 + 2 | 2 (2 times) | double transposition: two cycles of size two | ![]() ![]() ![]() |
3 | ![]() |
even; no | 2 | ![]() |
3 + 1 | 3 (1 time), 1 (1 time) | one 3-cycle, one fixed point | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | ![]() ![]() |
even; yes; no | 3 | ![]() |
4 | 4 (1 time) | one 4-cycle, no fixed points | ![]() ![]() ![]() ![]() ![]() ![]() |
6 | ![]() ![]() |
odd | 4 | ![]() |
Total (5 rows, 5 being the number of unordered integer partitions of 4) | -- | -- | -- | 24 (equals 4!, the order of the whole group) | -- | odd: 12 (2 classes) even; no: 4 (2 classes) even; yes; no: 8 (1 class) |
order 1: 1 (1 class) order 2: 9 (2 classes) order 3: 8 (1 class) order 4: 6 (1 class) |
-- |
Here is more information on the conjugacy classes:
FACTS TO CHECK AGAINST ON FIXED POINTS AND CYCLES
Fixed points: probability distribution of number of fixed points of permutations | expected number of fixed points of permutation equals one
Number of cycles: probability distribution of number of cycles of permutations | expected number of cycles of permutation equals harmonic number of degree
Partition | Number of elements in conjugacy class | Order of elements | Number of fixed points | Number of cycles (including fixed points) | Minimum number of transpositions that must be multiplied to obtain this cycle decomposition |
---|---|---|---|---|---|
1 + 1 + 1 + 1 | 1 | 1 | 4 | 4 | 0 |
2 + 1 + 1 | 6 | 2 | 2 | 3 | 1 |
2 + 2 | 3 | 2 | 0 | 2 | 2 |
3 + 1 | 8 | 3 | 1 | 2 | 2 |
4 | 6 | 4 | 0 | 1 | 3 |
Mean over conjugacy classes | 24/5 | 8/5 | 7/5 | 6 | 8/5 |
Mean over elements | 73/12 | 67/24 | 1 | 25/12 | 23/12 |
The mean over elements of the number of fixed points is for all symmetric groups on finite sets. The mean over elements of the number of cycles is
, which in this case is
.
For characters, see linear representation theory of symmetric group:S4.
Interpretation as projective general linear group of degree two
The symmetric group is isomorphic to
, i.e., the projective general linear group of degree two over field:F3. Compare with element structure of projective general linear group of degree two over a finite field.
Nature of conjugacy class upstairs in ![]() ![]() |
Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class (generic odd ![]() |
Size of conjugacy class (![]() |
Number of such conjugacy classes (generic odd ![]() |
Number of such conjugacy classes (![]() |
Total number of elements (generic odd ![]() |
Total number of elements (![]() |
Representatives of conjugacy classes as permutations |
---|---|---|---|---|---|---|---|---|---|---|
Diagonalizable over ![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() |
1 | 1 | 1 | 1 | 1 | 1 | ![]() |
Diagonalizable over ![]() ![]() |
Pair of mutually negative conjugate elements of ![]() |
![]() ![]() |
Same as characteristic polynomial | ![]() |
3 | 1 | 1 | ![]() |
3 | ![]() |
Diagonalizable over ![]() |
![]() |
![]() |
Same as characteristic polynomial | ![]() ![]() |
6 | 1 | 1 | ![]() ![]() |
6 | ![]() |
Diagonalizable over ![]() ![]() |
Pair of conjugate elements of ![]() ![]() |
![]() ![]() |
Same as characteristic polynomial | ![]() |
6 | ![]() |
1 | ![]() ![]() |
6 | ![]() |
Not diagonal, has Jordan block of size two | ![]() |
![]() |
Same as characteristic polynomial | ![]() |
8 | 1 | 1 | ![]() |
8 | ![]() |
Diagonalizable over ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() |
Same as characteristic polynomial. | ![]() |
12 | ![]() |
0 | ![]() ![]() |
0 | -- |
Total | NA | NA | NA | NA | NA | ![]() |
5 | ![]() |
24 | -- |
Interpretation as general affine group of degree two
The symmetric group is isomorphic to
, i.e., the general affine group of degree two over field:F2. Compare with element structure of general affine group of degree two over a finite field. In the table below,
. The transformation is of the form
where
and
.
Nature of conjugacy class | Eigenvalues | Characteristic polynomial of ![]() |
Minimal polynomial of ![]() |
Size of conjugacy class (generic ![]() |
Size of conjugacy class (![]() |
Number of such conjugacy classes (generic ![]() |
Number of such conjugacy classes (![]() |
Total number of elements (generic ![]() |
Total number of elements (![]() |
Representative element as permutation, one for each conjugacy class |
---|---|---|---|---|---|---|---|---|---|---|
![]() ![]() |
![]() |
![]() |
![]() |
1 | 1 | 1 | 1 | 1 | 1 | ![]() |
![]() ![]() |
![]() |
![]() |
![]() |
![]() |
3 | 1 | 1 | ![]() |
3 | ![]() |
![]() ![]() ![]() |
![]() ![]() |
![]() ![]() |
![]() ![]() |
![]() |
4 | ![]() |
0 | ![]() |
0 | -- |
![]() ![]() ![]() ![]() |
Pair of conjugate elements of ![]() |
![]() |
Same as characteristic polynomial | ![]() |
8 | ![]() |
1 | ![]() |
8 | ![]() |
![]() ![]() ![]() |
![]() |
![]() |
Same as characteristic polynomial | ![]() |
6 | 1 | 1 | ![]() |
6 | ![]() |
![]() ![]() ![]() |
![]() |
![]() |
Same as characteristic polynomial | ![]() |
6 | 1 | 1 | ![]() |
6 | ![]() |
![]() |
![]() ![]() |
![]() ![]() |
Same as characteristic polynomial | ![]() |
12 | ![]() |
0 | ![]() |
0 | -- |
![]() ![]() ![]() ![]() |
![]() ![]() |
![]() |
Same as characteristic polynomial | ![]() |
12 | ![]() |
0 | ![]() |
0 | -- |
![]() ![]() ![]() ![]() |
![]() ![]() |
![]() |
Same as characteristic polynomial | ![]() |
12 | ![]() |
0 | ![]() |
0 | -- |
![]() ![]() |
![]() ![]() |
![]() |
Same as characteristic polynomial | ![]() |
24 | ![]() |
0 | ![]() |
0 | -- |
Total | NA | NA | NA | NA | NA | ![]() |
5 | ![]() |
24 |
Interpretation as general semiaffine group of degree one
Compare and contrast with element structure of general semiaffine group of degree one over a finite field
We view the group as the general semiaffine group of degree one with
. Here,
and
.
Nature of conjugacy class | Size of conjugacy class (generic ![]() |
Size of conjugacy class (![]() |
Number of such conjugacy classes (generic ![]() |
Number of such conjugacy classes (![]() |
Total number of elements (generic ![]() |
Total number of elements (![]() |
Representative as permutation (one per class) |
---|---|---|---|---|---|---|---|
identity element of group, acts as identity map | 1 | 1 | 1 | 1 | 1 | 1 | ![]() |
non-identity element, in additive group. Acts as translation | ![]() |
3 | 1 | 1 | ![]() |
3 | ![]() |
outside the additive group, but in GA(1,q) and the multiplicative part is in the prime subfield. Acts as ![]() ![]() |
![]() |
4 | ![]() |
0 | ![]() |
0 | -- |
outside the additive group, but in GA(1,q) and the multiplicative part is outside the prime subfield. Acts as ![]() ![]() |
![]() |
8 | ![]() |
1 | ![]() |
8 | ![]() |
outside GA(1,q), has order two (i.e., generates a permutable complement to the subgroup of index two that is GA(1,q)) | ![]() |
6 | 1 | 1 | ![]() |
6 | ![]() |
outside GA(1,q), its image mod the additive group has order two but it does not itself have order two | ![]() |
6 | 1 | 1 | ![]() |
6 | ![]() |
outside GA(1,q), its image mod the additive group does not have order two | ![]() |
12 | ![]() |
0 | ![]() |
0 | -- |
Total | -- | -- | ![]() |
5 | ![]() |
24 | -- |
Conjugacy class structure: additional information
Number of conjugacy classes
The symmetric group of degree four has 5 conjugacy classes. Below are listed various methods that can be used to compute the number of conjugacy classes, all of which should give the answer 5:
Convolution algebra on conjugacy classes
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Cayley graphs
With generating set all transpositions
Here, the generating set is the set of all transpositions. Since the generating set is a conjugacy class of involutions, the left and right Cayley graphs are identical. Further, we can unambiguously give a direction to each edge (away from the identity element) because there are no cycles of odd length, which follows from the fact that all elements of the generating set are odd permutations.
The following is some useful tabulated information about the Cayley graph. The edges to/from listed here are the edges for any representative element, not the total across the conjugacy class. Note that the sum of edges to/from in each row is , which is the number of generators used in the generating set.
Conjugacy class | Distance from identity in Cayley graph | Number of elements | Edges to/from ![]() |
Edges to/from ![]() |
Edges to/from ![]() |
Edges to/from ![]() |
Edges to/from ![]() |
---|---|---|---|---|---|---|---|
![]() |
0 | 1 | 0 | 6 | 0 | 0 | 0 |
![]() |
1 | 6 | 1 | 0 | 4 | 1 | 0 |
![]() |
2 | 8 | 0 | 3 | 0 | 0 | 3 |
![]() |
2 | 3 | 0 | 2 | 0 | 0 | 4 |
![]() |
3 | 6 | 0 | 0 | 4 | 2 | 0 |
Bruhat ordering
The basic picture:
A fuller picture:
Basic tabulation by length
Length | Number of elements of that length | Elements of that length | Conjugacy class information for these elements |
---|---|---|---|
0 | 1 | ![]() |
forms a single conjugacy class |
1 | 3 | ![]() ![]() ![]() |
3 of 6 elements in conjugacy class of ![]() |
2 | 5 | ![]() ![]() ![]() ![]() ![]() |
1 of 3 elements in conjugacy class of ![]() ![]() |
3 | 6 | ![]() ![]() ![]() ![]() ![]() ![]() |
2 of 6 elements in conjugacy class of ![]() ![]() |
4 | 5 | ![]() ![]() ![]() ![]() ![]() |
1 of 3 elements in conjugacy class of ![]() ![]() |
5 | 3 | ![]() ![]() ![]() |
2 of 6 elements in conjugacy class of ![]() ![]() |
6 | 1 | ![]() |
1 of 3 elements in conjugacy class of ![]() |
Equivalence classes up to symmetries
These are equivalence classes up to the two symmetries: flipping the s and a left-to-right order reversal symmetry. Elements in the same equivalence class are in the same conjugacy class in the group and the corresponding points in the Bruhat ordering are in the same orbit under automorphisms of the graph of the Bruhat ordering.
Elements in equivalence class | Number of elements | Length | Indegree | Outdegree | Image class under the anti-automorphism |
---|---|---|---|---|---|
![]() |
1 | 0 | 0 | 1 | ![]() |
![]() ![]() |
2 | 1 | 1 | 3 | ![]() ![]() |
![]() |
1 | 1 | 1 | 4 | ![]() |
![]() ![]() ![]() ![]() |
4 | 2 | 2 | 3 | ![]() ![]() ![]() ![]() |
![]() |
1 | 2 | 2 | 2 | ![]() |
![]() ![]() |
2 | 3 | 2 | 2 | ![]() ![]() |
![]() ![]() |
2 | 3 | 2 | 2 | ![]() ![]() |
![]() ![]() |
2 | 3 | 3 | 3 | ![]() ![]() |
![]() ![]() ![]() ![]() |
4 | 4 | 3 | 2 | ![]() ![]() ![]() ![]() |
![]() |
1 | 4 | 2 | 2 | ![]() |
![]() ![]() |
2 | 5 | 3 | 1 | ![]() ![]() |
![]() |
1 | 5 | 4 | 1 | ![]() |
![]() |
1 | 6 | 1 | 0 | ![]() |