Element structure of symmetric group:S3
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This article gives specific information, namely, element structure, about a particular group, namely: symmetric group:S3.
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This article discusses symmetric group:S3, the symmetric group of degree three. We denote its elements as acting on the set , written using cycle decompositions, with composition by function composition where functions act on the left. The multiplication table is given below. The convention followed here is that the row element is multiplied on the left and the column element is multiplied on the right. Since functions are assumed to act on the left, this implies that the column element is the permutation that operates first:
Element | ||||||
---|---|---|---|---|---|---|
If we assume functions to act on the right, then the multiplication table constructed must be interpreted taking the row element as the element multiplied on the right and the column element as the element multiplied on the left.
For a complete explanation of how this multiplication table can be constructed, see the survey article construction of multiplication table of symmetric group:S3.
This article focuses on the basic abstract group structure and key attributes. For more on the combinatorics that arises specifically from its being a symmetric group, see combinatorics of symmetric group:S3.
Contents
Summary
Item | Value |
---|---|
order of the whole group (total number of elements) | 6 See element structure of symmetric group:S3#Order computation |
conjugacy class sizes | 1,2,3 maximum: 3, number of conjugacy classes: 3, lcm: 6 |
number of conjugacy classes | 3 See element structure of symmetric group:S3#Number of conjugacy classes |
order statistics | 1 of order 1, 3 of order 2, 2 of order 3 maximum: 3, lcm (exponent of the whole group): 6 |
Family contexts
Family name | Parameter values | Information on element structure of family |
---|---|---|
symmetric group of degree | degree , i.e., | element structure of symmetric groups |
general linear group of degree two over a finite field of size a prime power . | prime power , i.e., field:F2: The group is | element structure of general linear group of degree two over a finite field |
dihedral group of degree , order | degree , order , i.e., the group | element structure of dihedral groups |
general affine group of degree one over a finite field of size a prime power . | prime power , i.e., field:F3, so the group is . | element structure of general affine group of degree one over a finite field |
Note: By isomorphism between linear groups over field:F2, we obtain that all the groups , , , and are isomorphic to each other, and hence to . Hence, we can also study in terms of element structure of projective general linear group of degree two over a finite field, element structure of special linear group of degree two over a finite field, and element structure of projective special linear group of degree two over a finite field.
Elements
Multiple ways of describing permutations
Cycle decomposition notation (cycles for fixed points are omitted) | One-line notation, i.e., image of string | Order | Matrix (left action convention) | Matrix (right action convention) | Comment |
---|---|---|---|---|---|
123 | 1 | The matrices are the same for the identity element. | |||
213 | 2 | The matrices are the same, since the element has order 2. | |||
132 | 2 | The matrices are the same, since the element has order 2. | |||
321 | 2 | The matrices are the same, since the element has order 2. | |||
231 | 3 | The matrices are transposes of each other, and are not equal to each other, since the element does not have order 2. | |||
312 | 3 | The matrices are transposes of each other, and are not equal to each other, since the element does not have order 2. |
Here is the multiplication table using the one-line notation:
Element | 123 | 213 | 132 | 321 | 231 | 312 |
---|---|---|---|---|---|---|
123 | 123 | 213 | 132 | 321 | 231 | 312 |
213 | 213 | 123 | 231 | 312 | 132 | 321 |
132 | 132 | 312 | 123 | 231 | 321 | 213 |
321 | 321 | 231 | 312 | 123 | 213 | 132 |
231 | 231 | 321 | 213 | 132 | 312 | 123 |
312 | 312 | 132 | 321 | 213 | 123 | 231 |
Order computation
The symmetric group of degree three has order 6. Below are listed various methods that can be used to compute the order, all of which should give the answer 6:
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 dividing the order | Case (answer should be 1) | Case (answer should be 1) | Case (answer should be 0) |
---|---|---|---|---|---|---|
symmetric group of degree | degree | no summands, so sum is 0 | ||||
projective general linear group of degree two over a finite field of size | size , i.e., field:F2, so the group is | In factored form: |
If is the underlying prime of , then . If and , then 1 + the exponent for the largest power of 2 dividing . If and , then 1 + the exponent for the largest power of 2 dividing . Otherwise, if divides or (can't divide both) the exponent for the largest power dividing that. If it divides neither, then zero. |
Here, is the underlying prime of , so . | This is the case divides , and the power is 1. | Since 2 - 1 = 1 and 2 + 1 = 3 are powers of 2, there are no primes involved other than 2 and 3. |
dihedral group of degree , order | degree , order | largest power of dividing . | 1 | 1 | no higher prime divides 6. | |
general affine group of degree one over a finite field of size | , i.e., field:F3. | if is the underlying prime of , . Else, if divides , the exponent for the largest power dividing that. Otherwise, 0. | divides , the exponent is 1. | is the underlying prime of , so . | is neither the underlying prime of nor does it divide . |
Other operations induced by group multiplication
Self-action by conjugation
Below is the induced binary operation where the column element acts on the row element by conjugation on the left, i.e., if the row element is and the column element is , the cell is filled with .
Note that the action by conjugation functions by relabeling, so conjugating an element by an element effectively replaces each element in each cycle of the cycle decomposition of by the image of that element under .
Commutator operation
Here, the two inputs are group elements , and the output is the commutator. We first give the table assuming the left definition of commutator: . Here, the row element is and the column element is . Note that :
Here is the information on the number of times each element occurs as a commutator:
Conjugacy class (indexing partition) | Elements | Number of occurrences of each as commutator | Probability of each occurring as the commutator of elements picked uniformly at random | Total number of occurrences as commutator | Total probability | Explanation |
---|---|---|---|---|---|---|
1 + 1 + 1 | 18 | 1/2 | 18 | 1/2 | See commuting fraction and its relationship with the number of conjugacy classes. | |
2 + 1 | 0 | 0 | 0 | 0 | Not in the derived subgroup. | |
3 | 9 | 1/4 | 18 | 1/2 |
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 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 in cycle decomposition notation | Elements with the cycle type in one-line notation | 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 (3 times) | three fixed points | -- the identity element | 123 | 1 | even; no | 1 | ||
2 + 1 | 2 (1 time), 1 (1 time) | transposition in symmetric group:S3: one 2-cycle, one fixed point | , , | 213, 321, 132 | 3 | odd | 2 | ||
3 | 3 (1 time) | 3-cycle in symmetric group:S3: one 3-cycle | , | 231, 312 | 2 | even; yes; no | 3 | ||
Total (3 rows -- 3 being the number of unordered integer partitions of 3) | -- | -- | -- | -- | 6 (equals 3!, the size of the symmetric group) | -- | odd: 3 even;no: 1 even; yes; no: 2 |
order 1: 1, order 2: 3, order 3: 2 | -- |
This group is one of three finite groups with the property that any two elements of the same order are conjugate. The other two are the cyclic group of order two and the trivial group.
Here is some more information:
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 | 3 | 3 | 0 |
2 + 1 | 3 | 2 | 1 | 2 | 1 |
3 | 2 | 3 | 0 | 1 | 2 |
Mean over conjugacy classes | 2 | 2 | 4/3 | 2 | 1 |
Mean over elements | 7/3 | 13/6 | 1 | 11/6 | 7/6 |
Note that the mean over elements of the number of fixed points is 1 for any symmetric group on a finite set, and the average of the number of cycles is .
For characters, see linear representation theory of symmetric group:S3.
Interpretation as general linear group of degree two
Compare with element structure of general linear group of degree two#Conjugacy class structure
This group is the general linear group of degree two over field:F2.
Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class (generic ) | Size of conjugacy class () | Number of such conjugacy classes (generic ) | Number of conjugacy classes () | Total number of elements (generic ) | Total number of elements () | Representative as permutation |
---|---|---|---|---|---|---|---|---|---|---|
Diagonalizable over (here ) with equal diagonal entries, hence a scalar | 1 | 1 | 1 | 1 | ||||||
Diagonalizable over (here ), not over (here ). Must necessarily have no repeated eigenvalues. | Pair of conjugate elements of | , irreducible | Same as characteristic polynomial | 2 | 1 | 2 | -- see 3-cycle in symmetric group:S3 | |||
Not diagonal, has Jordan block of size two | (multiplicity two) | Same as characteristic polynomial | 3 | 1 | 3 | -- see transposition in symmetric group:S3 | ||||
Diagonalizable over with distinct diagonal entries | -- | -- | -- | 6 | 0 | 0 | -- | |||
Total (--) | -- | -- | -- | -- | -- | 3 | 6 | -- |
Interpretation as dihedral group
Compare with element structure of dihedral groups#Odd degree case
The symmetric group of degree three is isomorphic to the dihedral group of degree three and order six (i.e., it is the dihedral group of order where ). In the table below, we denote by the generator of the cyclic subgroup of order three (which we could take as the permutation ) and by one of the reflections (which we could take as ).
Nature of conjugacy class | Size of each conjugacy class (generic odd ) | Size of each 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 as permutations |
---|---|---|---|---|---|---|---|
Identity element | 1 | 1 | 1 | 1 | 1 | 1 | |
Non-identity elements in cyclic subgroup , where each element and its inverse form a conjugacy class | 2 | 2 | 1 | 2 | -- see 3-cycle in symmetric group:S3 | ||
Elements outside the cyclic subgroup , all form a single conjugacy class | 3 | 1 | 1 | 3 | -- see transposition in symmetric group:S3 | ||
Total (--) | -- | -- | 3 | 6 | -- |
Interpretation as general affine group of degree one
Compare with element structure of general affine group of degree one over a finite field#Conjugacy class structure
The symmetric group of degree three is isomorphic to the general affine group of degree one over field:F3. All the elements of this group are of the form:
where . Below, we interpret the conjugacy classes of the group in these terms:
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 () | Representatives of conjugacy classes as permutations |
---|---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | 1 | 1 | ||
(conjugacy class is independent of choice of ) | 2 | 1 | 1 | 2 | -- see 3-cycle in symmetric group:S3 | ||
(conjugacy class is determined completely by choice of and is independent of choice of ; in other words, each conjugacy class is a coset of the subgroup of translations) | 3 | 1 | 3 | -- see transposition in symmetric group:S3 | |||
Total (--) | -- | -- | 3 | 6 | -- |
Interpretation as general semilinear group of degree one
Compare with element structure of general semilinear group of degree one over a finite field#Conjugacy class structure
The symmetric group of degree three is isomorphic to the general semilinear group of degree one over field:F4. In other words, it is the group for . We will denote alternately by .
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) |
---|---|---|---|---|---|---|---|
in the multiplicative group and in the prime subfield | 1 | 1 | 1 | 1 | |||
outside the prime subfield | 2 | 2 | 1 | 2 | -- see 3-cycle in symmetric group:S3 | ||
outside the multiplicative group | 3 | 1 | 3 | -- see transposition in symmetric group:S3 | |||
Total | -- | -- | (equals number of conjugacy classes in the group) | 3 | (equals order of the whole group) | 6 | -- |
Conjugacy class structure: additional information
Number of conjugacy classes
The symmetric group of degree three has 3 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 3:
Convolution algebra on conjugacy classes
The convolution algebra on conjugacy classes for this group is given by:
Partition/conjugacy class | |||
---|---|---|---|
Rational and real conjugacy classes
Since the symmetric group of degree three is a rational group and in particular an ambivalent group, the rational conjugacy classes coincide with the conjugacy classes and the real conjugacy classes also coincide with the conjugacy classes.
Further information: symmetric groups are rational
Action of automorphism group on conjugacy classes
Since the symmetric group of degree three is a complete group, i.e., every automorphism is inner, the automorphism group acts as the identity on the set of conjugacy classes.
Note that the symmetric group of degree for is complete. Further information: symmetric groups on finite sets are complete