Element structure of alternating group:A5
This article gives specific information, namely, element structure, about a particular group, namely: alternating group:A5.
View element structure of particular groups | View other specific information about alternating group:A5
This article gives the element structure of alternating group:A5.
See also element structure of alternating groups and element structure of symmetric group:S5.
Summary
Item | Value |
---|---|
order of the whole group (total number of elements) | 60 prime factorization See order computation for more |
conjugacy class sizes | 1,12,12,15,20 maximum: 20, number: 5, sum (equals order of group): 60, lcm: 60 See conjugacy class structure for more. |
number of conjugacy classes | 5 See element structure of alternating group:A5#Number of conjugacy classes |
order statistics | 1 of order 1, 15 of order 2, 20 of order 3, 24 of order 5 maximum: 5, lcm (exponent of the whole group): 30 |
Family contexts
Family name | Parameter values | General discussion of element structure of family |
---|---|---|
alternating group | 5 | element structure of alternating groups |
projective general linear group of degree two over a finite field | field:F4 | element structure of projective general linear group of degree two over a finite field |
projective special linear group of degree two over a finite field | field:F5 | element structure of projective special linear group of degree two over a finite field |
COMPARE AND CONTRAST: View element structure of groups of order 60 to compare and contrast the element structure with other groups of order 60.
Elements
Order computation
The alternating group of degree five has order 60, with prime factorization . Below are listed various methods that can be used to compute the order, all of which should give the answer 60:
Family | Parameter values | Formula for order of a group in the family | Proof or justification of formula | Evaluation at parameter values | Full interpretation of conjugacy class structure | Orders for members of this family |
---|---|---|---|---|---|---|
alternating group of degree | degree | See alternating group, element structure of alternating groups | #Interpretation as alternating group | For : 3,12,60,360,2520,20160,... | ||
projective special linear group of degree two over a finite field of size | , i.e., field:F5, so the group is | which becomes , same as for odd , same as for a power of 2 |
See order formulas for linear groups of degree two, order formulas for linear groups, and projective special linear group of degree two | Factored version: |
#Interpretation as projective special linear group of degree two | For (odd case): 12,60,168,360,... |
special linear group of degree two over a finite field of size | , i.e., field:F4, so the group is | , same as | See order formulas for linear groups of degree two, order formulas for linear groups, and special linear group of degree two | In factored form: |
#Interpretation as special linear group of degree two over field:F4 | For : 6,24,60,120,336,504,720,... |
von Dyck group with parameters | (note that the order of the parameters is irrelevant, though we usually arrange them in ascending or descending order depending on the convention being followed). | See element structure of von Dyck groups | #Interpretation as von Dyck group | (omitted here due to multiple parameters) |
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
There is a total of 5 conjugacy classes, of which 3 are unsplit from symmetric group:S5, and 2 are a split pair arising from a single conjugacy class in . The conjugacy class sizes are 1, 12, 12, 15, 20.
Interpretation as alternating 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 a symmetric group, cycle type determines conjugacy class. The statement is almost true for the alternating group, except for the fact that some conjugacy classes of even permutations in the symmetric group split into two in the alternating group, as per the splitting criterion for conjugacy classes in the alternating group, which says that a conjugacy class of even permutations splits in the alternating group if and only if it is the product of odd cycles of distinct length.
Here are the unsplit conjugacy classes:
Partition | Verbal description of cycle type | Representative element of the cycle type | All elements of the cycle type | Size of conjugacy class | Formula for size | Element order |
---|---|---|---|---|---|---|
1 + 1 + 1 + 1 + 1 | five fixed points | -- the identity element | 1 | 1 | ||
3 + 1 + 1 | one 3-cycle, two fixed points | [SHOW MORE] | 20 | 3 | ||
2 + 2 + 1 | double transposition: two 2-cycles, one fixed point | [SHOW MORE] | 15 | 2 |
Here is the split pair of conjugacy classes:
Partition | Verbal description of cycle type | Combined size of conjugacy classes | Formula for combined size | Size of each half | Representative of first half | Representative of second half | Real? | Rational? | Element order |
---|---|---|---|---|---|---|---|---|---|
5 | one 5-cycle | 24 | 12 | Yes | No | 5 |
Interpretation as projective special linear group of degree two
Compare with element structure of projective special linear group of degree two over a finite field#Conjugacy class structure.
We consider the group as with . We use the letter to denote the generic case of .
Nature of conjugacy class upstairs in | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class (generic that is 1 mod 4) | Size of conjugacy class () | Number of such conjugacy classes (generic that is 1 mod 4) | Number of such conjugacy classes () | Total number of elements (generic that is 1 mod 4) | Total number of elements () | Matrix representatives upstairs (one per conjugacy class) | Representatives as permutations |
---|---|---|---|---|---|---|---|---|---|---|---|
Diagonalizable over with equal diagonal entries, hence a scalar | or , both correspond to the same element | where | where | 1 | 1 | 1 | 1 | 1 | 1 | ||
Not diagonal, has Jordan block of size two | (multiplicity 2) or (multiplicity 2). Each conjugacy class has one representative of each type. | where | Same as characteristic polynomial | 12 | 2 | 2 | 24 | , | , | ||
Diagonlizable over with diagonal entries squaring to | 15 | 1 | 1 | 15 | |||||||
Diagonalizable over , not over . Must necessarily have no repeated eigenvalues. | Pair of conjugate elements of field:F25 of norm 1. Each pair identified with its negative pair. | , , get identified. | Same as characteristic polynomial | 20 | 1 | 20 | |||||
Diagonalizable over with distinct (and hence mutually inverse) diagonal entries, whose square is not | None for this field | -- | -- | 30 | 0 | 0 | -- | -- | |||
Total | NA | NA | NA | NA | NA | 5 | 60 | NA | NA |
Interpretation as special linear group of degree two over field:F4
Compare with element structure of special linear group of degree two#Conjugacy class structure.
Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Semisimple? | Diagonalizable over ? | Splits in relative to ? | Representative matrices (one for each conjugacy class) | Representative element as permutation |
---|---|---|---|---|---|---|---|---|---|---|---|
Diagonalizable over field:F4 with distinct (and hence mutually inverse) diagonal entries | where | 20 | 1 | 20 | Yes | Yes | No | ||||
Diagonalizable over field:F16, not over field:F4. Must necessarily have no repeated eigenvalues. | Pair of conjugate elements of field:F16 of norm 1 | , . | Same as characteristic polynomial | 12 | 2 | 24 | Yes | No | No | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | , |
Diagonalizable over field:F4 with equal diagonal entries, hence a scalar. | 1 | 1 | 1 | Yes | Yes | No | |||||
Not diagonal, has Jordan block of size two | (multiplicity 2) | 15 | 1 | 15 | No | No | No | ||||
Total | NA | NA | NA | NA | 5 | 60 | 45 | NA | NA |
Conjugacy class structure: additional information
Number of conjugacy classes
The alternating group of degree five 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: