# Element structure of alternating group:A6

This article gives specific information, namely, element structure, about a particular group, namely: alternating group:A6.

View element structure of particular groups | View other specific information about alternating group:A6

This article gives the element structure of alternating group:A6.

See also element structure of alternating groups and element structure of symmetric group:S5.

For convenience, we take the set acted upon as .

## Elements

### Order computation

The alternating group of degree six has order 360, with prime factorization . Below are listed various methods that can be used to compute the order, all of which should give the answer 360:

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 |
---|---|---|---|---|---|

alternating group of degree | degree | See alternating group, element structure of alternating groups | #Interpretation as alternating group | ||

projective special linear group of degree two over a finite field of size | , i.e., field:F9, so the group is | for odd 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 |

## 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 centralizerBounding facts: size of conjugacy class is bounded by order of derived subgroupCounting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

There is a total of 7 conjugacy classes, of which 5 are unsplit from symmetric group:S6, and 2 are a split pair arising from a single conjugacy class in . The conjugacy class sizes are 1, 40, 45, 90, 40, 72, 72.

### 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 groupOther 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 | Size of conjugacy class | Formula for size | Element order |
---|---|---|---|---|---|

1 + 1 + 1 + 1 + 1 + 1 | six fixed points | -- the identity element | 1 | 1 | |

3 + 1 + 1 + 1 | one 3-cycle, three fixed points | 40 | 3 | ||

2 + 2 + 1 + 1 | double transposition: two 2-cycles, two fixed points | 45 | 2 | ||

4 + 2 | one 4-cycle, one 2-cycle | 90 | 4 | ||

3 + 3 | two 3-cycles | 40 | 3 |

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 + 1 | one 5-cycle, one fixed point | 144 | 72 | 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 , . 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 | where | 40 | 2 | 2 | 80 | and | |||

Diagonalizable over with diagonal entries squaring to | where | 45 | 1 | 1 | 45 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
|||||

Diagonalizable over , not over . Must necessarily have no repeated eigenvalues. | Pair of conjugate elements of of norm 1. Each pair identified with its negative pair. | , irreducible; note that 's pair and 's pair get identified. | Same as characteristic polynomial | 72 | 2 | 144 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
and | |||

Diagonalizable over with distinct (and hence mutually inverse) diagonal entries, whose square is not |
where where are square roots of . Note that the representative pairs and get identified. | , again with identification. | , again with identification. | 90 | 1 | 90 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
||||

Total | NA | NA | NA | NA | NA | 7 | 360 | NA | NA |

## Conjugacy class structure: additional information

### Number of conjugacy classes

The alternating group of degree six has 7 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 7:

Family | Parameter values | Formula for number of conjugacy classes of a group in the family | Proof or justification of formula | Evaluation at parameter values | Full interpretation of conjugacy class structure |
---|---|---|---|---|---|

alternating group of degree | , i.e., the group | (Number of pairs of non-self-conjugate partitions of ) + 2(Number of self-conjugate partitions of ) | See element structure of alternating groups | #Interpretation as alternating group | |

projective special linear group of degree two over a finite field of size | , i.e., field:F9, so the group is | for odd for a power of 2 |
See element structure of projective special linear group of degree two over a finite field, number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size | #Interpretation as projective special linear group of degree two |