# Endomorphism structure of symmetric groups

From Groupprops

This article gives specific information, namely, endomorphism structure, about a family of groups, namely: symmetric group.

View endomorphism structure of group families | View other specific information about symmetric group

(order of symmetric group) | symmetric group of degree | endomorphism structure page | |
---|---|---|---|

0 | 1 | trivial group | -- |

1 | 1 | trivial group | -- |

2 | 2 | cyclic group:Z2 | -- |

3 | 6 | symmetric group:S3 | endomorphism structure of symmetric group:S3 |

4 | 24 | symmetric group:S4 | endomorphism structure of symmetric group:S4 |

5 | 120 | symmetric group:S5 | endomorphism structure of symmetric group:S5 |

6 | 720 | symmetric group:S6 | endomorphism structure of symmetric group:S6 |

## Endomorphism structure

### Automorphism structure

- For , the symmetric group of degree is a complete group, i.e., the action on itself by conjugation gives an isomorphism from the group to its automorphism group. In particular, the symmetric group is isomorphic to its automorphism group.
- For , the automorphism group is trivial.
- For , i.e., for symmetric group:S6, the action on itself by conjugation induces an
*injective*homomorphism to the automorphism group that is not surjective. The image has index two in the automorphism group. In other words, the outer automorphism group is cyclic group:Z2. See automorphism group of symmetric group:S6.

### Other endomorphisms

- For , every endomorphism is an automorphism.
- For , every endomorphism is either trivial or an automorphism.
- For and , every endomorphism is of one of these three types: an automorphism, the trivial map, or a map with image a cyclic subgroup of order two, obtained by applying the sign homomorphism. For the last type of endomorphism, if the non-identity element in the image is an odd permutation, then the map is a retraction, whereas if it is an even permutation, it is not a retraction.
- For , there are additional endomorphisms possible due to the presence of a normal V4 in S4.

Thus, for a generic , we have:

Quantity | Value |
---|---|

order of automorphism group | (it's the symmetric group itself) |

order of endomorphism monoid | plus the number of elements of of order two. |

number of retractions | plus the number of odd permutations of of order two. |

### Summary information

(order of symmetric group) | symmetric group of degree | number of odd permutations of order two | total number of permutations of order two | automorphism group (= symmetric group of degree except when ) | order of automorphism group (= except when ) | order of outer automorphism group (= 1 except when ) | order of endomorphism monoid (= order of automorphism group + total number of permutations of order two + 1, except when ) | number of retractions (= 2 + number of odd permutations of order two, except when ) | |
---|---|---|---|---|---|---|---|---|---|

0 | 1 | trivial group | 0 | 0 | trivial group | 1 | 1 | 1 | 1 |

1 | 1 | trivial group | 0 | 0 | trivial group | 1 | 1 | 1 | 1 |

2 | 2 | cyclic group:Z2 | 1 | 1 | trivial group | 1 | 1 | 2 | 2 |

3 | 6 | symmetric group:S3 | 3 | 3 | symmetric group:S3 | 6 | 1 | 10 | 5 |

4 | 24 | symmetric group:S4 | 6 | 9 | symmetric group:S4 | 24 | 1 | 58 | 12 |

5 | 120 | symmetric group:S5 | 10 | 25 | symmetric group:S5 | 120 | 1 | 146 | 12 |

6 | 720 | symmetric group:S6 | 30 | 75 | automorphism group of symmetric group:S6 | 1440 | 2 | 1516 | 32 |

7 | 5040 | symmetric group:S7 | 126 | 231 | symmetric group:S7 | 5040 | 1 | 5272 | 128 |