# Alternating group:A4

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## Contents

## Definition

The alternating group is defined in the following equivalent ways:

- It is the group of even permutations (viz., the alternating group) on four elements.
- It is the von Dyck group (sometimes termed
*triangle group*, though triangle group has an alternative interpretation) with parameters (sometimes written in reverse order as ). - It is the group of orientation-preserving symmetries of a regular tetrahedron. When viewed in this light, it is called the
**tetrahedral group**, and its symbol as a point group is or .`Further information: Classification of finite subgroups of SO(3,R), Linear representation theory of alternating group:A4` - It is the projective special linear group of degree two over the field of three elements, viz., .
- It is the general affine group of degree over the field of four elements, viz., (also written as .

### Equivalence of definitions

- von Dyck group with parameters (2,3,3) is isomorphic to A4
- Tetrahedral group is isomorphic to A4
- PSL(2,3) is isomorphic to A4
- GA(1,4) is isomorphic to A4

IMPORTANT NOTE: This page concentrates on as an abstract group in its own right. To learn more about this group as a subgroup of index two inside symmetric group:S4, see A4 in S4.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 12#Arithmetic functions

### Basic arithmetic functions

### Arithmetic functions of a counting nature

## Group properties

## Endomorphisms

### Automorphisms

The automorphism group of the alternating group of degree four is isomorphic to the symmetric group of degree four. Since the alternating group of degree four is centerless, it embeds as a subgroup inside its automorphism group.

Another way of thinking of this is as follows: in the symmetric group of degree four, the alternating group of degree four is a subgroup of index two, and every automorphism of this subgroup is realized as the restriction to this subgroup of an inner automorphism of the symmetric group.

### Endomorphisms

The endomorphisms of the alternating group of degree four are given by the following (i.e., equivalent to one of these up to composition with an automorphism):

- The trivial map.
- The identity map.
- The retraction to a subgroup of order three, with kernel being the Klein four-group comprising the identity and the double transpositions. (All such retractions are equivalent).

## Elements

`Further information: element structure of alternating group:A4`

### Summary

Item | Value |
---|---|

order of the whole group (total number of elements) | 12 (see order computation for more) |

conjugacy class sizes | 1,3,4,4 maximum: 4, number: 4, sum (equals order of whole group): 12, lcm: 12 See conjugacy class structure for more. |

number of conjugacy classes | 4 See number of conjugacy classes for more. |

order statistics | 1 of order 1, 3 of order 2, 8 of order 3 maximum: 3, lcm (exponent of the whole group): 6 |

### Up to conjugacy

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 its cycle decomposition comprises odd cycles of distinct length.

Here are the unsplit conjugacy classes:

Partition | Verbal description of cycle type | Elements with the cycle type | Size of conjugacy class | Formula for size | Element order |
---|---|---|---|---|---|

1 + 1 + 1 + 1 | four cycles of size one each, i.e., four fixed points | -- the identity element | 1 | 1 | |

2 + 2 | double transposition: two cycles of size two | , , | 3 | 2 | |

Total | -- | , , and | 4 | NA | NA |

In this case, the union of the unsplit conjugacy classes is a proper normal subgroup isomorphic to the Klein four-group. Note that this phenomenon is unique to the case .

Here is the split conjugacy class:

Partition | Verbal description of cycle type | Elements with the cycle type | Combined size of conjugacy classes | Formula for combined size | Size of each half | First split half | Second split half | Real? | Rational? | Element order |
---|---|---|---|---|---|---|---|---|---|---|

3 + 1 | one 3-cycle, one fixed point | , , , , , , , | 8 | 4 | , , , | , , , | No | No | 3 |

### Up to automorphism

The conjugacy classes of the identity element and double transpositions are invariant under all automorphisms.

An outer automorphism interchanges the conjugacy classes of elements of order three (each of size four). This can be realized, for instance, by viewing the alternating group as a subgroup of the symmetric group of degree four. Any transposition or -cycle in the symmetric group interchanges those two conjugacy classes

## Subgroups

`Further information: Subgroup structure of alternating group:A4`

### Quick summary

Item | Value |
---|---|

Number of subgroups | 10 |

Number of conjugacy classes of subgroups | 5 |

Number of automorphism classes of subgroups | 5 |

### Table classifying subgroups up to automorphism

Automorphism class of subgroups | List of subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes | Size of each conjugacy class | Total number of subgroups | Isomorphism class of quotient (if exists) | Subnormal depth (if subnormal) | Note |
---|---|---|---|---|---|---|---|---|---|---|

trivial subgroup | trivial group | 1 | 12 | 1 | 1 | 1 | alternating group:A4 | 1 | trivial | |

subgroup generated by double transposition in A4 | , , | cyclic group:Z2 | 2 | 6 | 1 | 3 | 3 | -- | 2 | |

V4 in A4 | Klein four-group | 4 | 3 | 1 | 1 | 1 | cyclic group:Z3 | 1 | 2-Sylow, minimal normal, maximal | |

A3 in A4 | , , , | cyclic group:Z3 | 3 | 4 | 1 | 4 | 4 | -- | -- | 3-Sylow, maximal |

whole group | all elements | alternating group:A4 | 12 | 1 | 1 | 1 | 1 | trivial group | 1 | whole |

Total (5 rows) | -- | -- | -- | -- | 5 | -- | 10 | -- | -- | -- |

## Supergroups

`Further information: supergroups of alternating group:A4`

### Subgroups: making all the automorphisms inner

`Further information: symmetric group:S4, A4 in S4`

The outer automorphism group of alternating group:A4 is cyclic group:Z2 and the automorphism group is symmetric group:S4. Since is centerless, it equals its inner automorphism group and hence embeds as a subgroup of index two inside symmetric group:S4.

In particular, symmetric group:S4 is the unique group containing alternating group:A4 as a NSCFN-subgroup (a normal fully normalized subgroup that is also a self-centralizing subgroup).

### Quotients: Schur covering groups

`Further information: group cohomology of alternating group:A4#Schur multiplier, second cohomology group for trivial group action of A4 on Z2`

`Further information: special linear group:SL(2,3), center of special linear group:SL(2,3)`

The Schur multiplier of alternating group:A4 is cyclic group:Z2. There is a *unique* corresponding Schur covering group, namely the group special linear group:SL(2,3), where the center of special linear group:SL(2,3) is isomorphic to the Schur multiplier cyclic group:Z2 and the quotient is alternating group:A4.

The Schur covering group is also denoted to indicate that it is a double cover of alternating group.

## Endomorphisms

`Further information: endomorphism structure of alternating group:A4`

Construct | Value | Order | Second part of GAP ID (if group) |
---|---|---|---|

endomorphism monoid | ? | 33 | -- |

automorphism group | symmetric group:S4 | 24 | 12 |

inner automorphism group | alternating group:A4 | 12 | 3 |

outer automorphism group | cyclic group:Z2 | 2 | 1 |

extended automorphism group | direct product of S4 and Z2 | 48 | 48 |

## Extensions

*These are groups having the alternating group as a quotient group*
Perhaps the most important of these is , which is the universal central extension of . The kernel of the projection mapping is a two-element subgroup, namely the identity matrix and the negative identity matrix.

## GAP implementation

### Group ID

This finite group has order 12 and has ID 3 among the groups of order 12 in GAP's SmallGroup library. For context, there are 5 groups of order 12. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(12,3)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(12,3);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [12,3]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other definitions

The alternating group can be constructed in many equivalent ways:

- As the alternating group of degree four, using GAP's AlternatingGroup function:

`AlternatingGroup(4)`

- Using the von Dyck presentation. Here is a sequence of steps:

F := FreeGroup(3); G := F/[F.1^3, F.2^3, F.3^2, F.1*F.2*F.3]

The output is the alternating group.

- As the projective special linear group, using GAP's ProjectiveSpecialLinearGroup function:

`PSL(2,3)`