# A3 in S5

From Groupprops

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z3 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

Let be the alternating group:A5, i.e., the alternating group (the group of all permutations) on the set . has order .

Consider the subgroup:

has a total of 10 conjugate subgroups (including itself) and the subgroups are parametrized by subsets of size 3 in describing the support of the 3-cycles. The complementary subset of size two is fixed point-wise by that conjugate subgroup:

Subset of size 3 | Complementary subset of size 2 fixed point-wise by the subgroup | Conjugate of |
---|---|---|

1,2,3 | 4,5 | |

1,2,4 | 3,5 | |

1,2,5 | 3,4 | |

1,3,4 | 2,5 | |

1,3,5 | 2,4 | |

1,4,5 | 2,3 | |

2,3,4 | 1,5 | |

2,3,5 | 1,4 | |

2,4,5 | 1,3 | |

3,4,5 | 1,2 |

Each of these subgroups is isomorphic to cyclic group:Z3.

## Arithmetic functions

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

order of the whole group | 120 | . See symmetric group:S5. |

[[order of a group|order of the subgroup | 3 | |

index of the subgroup | 40 | Follows from Lagrange's theorem |

size of conjugacy class of subgroups = index of normalizer | 10 | |

number of conjugacy classes in automorphism class | 1 |

## Effect of subgroup operators

In the table below, we provide values specific to .

Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|

normalizer | direct product of S3 and S2 in S5 | dihedral group:D12 | |

centralizer | Z6 in S5 | cyclic group:Z6 | |

normal core | trivial subgroup | -- | trivial group |

normal closure | the alternating subgroup | A5 in S5 | alternating group:A5 |

characteristic core | trivial subgroup | -- | trivial group |

characteristic closure | the alternating subgroup | A5 in S5 | alternating group:A5 |

## Conjugacy class-defining functions

Conjugacy class-defining function | What it means in general | Why it takes this value |
---|---|---|

Sylow subgroup for the prime | A -Sylow subgroup is a subgroup whose order is a power of and index is relatively prime to . Sylow subgroups exist and Sylow implies order-conjugate, i.e., all -Sylow subgroups are conjugate to each other. | The order of this subgroup is 3, which is the largest power of 3 dividing the order of the group. |