# Alternating group

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## Definition

### For a finite set

Let be a finite set. The **alternating group** on is defined in the following equivalent ways:

- It is the group of all even permutations on under composition. An even permutation is a permutation whose cycle decomposition has an even number of cycles of even size. Specifically, the alternating group on is the subgroup of the symmetric group on comprising the even permutations.
- It is the kernel of the sign homomorphism from the symmetric group on to the group .

For having size zero or one, the alternating group on equals the whole symmetric group on . For having size at least two, the alternating group on is the unique subgroup of index two in the symmetric group on .

The alternating group on a set of size is denoted and is termed the alternating group of degree .

### For an infinite set

Let be an infinite set. The **finitary alternating group** on is defined in the following equivalent ways:

- It is the group of all even permutations on under composition.
- It is the kernel of the sign homomorphism on the finitary symmetric group on .

## Facts

- The alternating group on a set of size five or more is simple. Also, the finitary alternating group on an infinite set is simple.
`For full proof, refer: A5 is simple, alternating groups are simple` - Projective special linear group equals alternating group in only finitely many cases

## Arithmetic functions

Here, is the degree of the alternating group, i.e., the size of the set it acts on.

For all the statements involving , we use the fact that A5 is simple and alternating groups are simple for degree at least five.

Function | Value | Explanation |
---|---|---|

order | for , for | It has index two in the symmetric group of degree . |

exponent | (skipping over ) if is odd, if is even. | |

nilpotency class | for , undefined for | abelian group for , nilpotent group for . |

derived length | for , for , undefined for | abelian for , simple for . |

Frattini length | for , for | Frattini-free group: intersection of maximal subgroups is trivial. |

Fitting length | for , for , undefined for | abelian for , simple for . |

minimum size of generating set | 2 | |

subgroup rank | At most | |

max-length | ? | |

chief length | for , for , for . | |

composition length | for , for , for . |

## Group properties

Property | Satisfied | Explanation |
---|---|---|

Abelian group | Yes for , no for | and don't commute. |

Nilpotent group | Yes for , no for | is centerless. |

Solvable group | Yes for , no for | alternating groups are simple for degree five or more. |

Supersolvable group | Yes for , no for | |

Simple group | Yes for and , no for . | |

Rational group | No for all . | |

Ambivalent group | Yes for , no otherwise | See classification of ambivalent alternating groups. |