# 2-Sylow subgroup of symmetric group

From Groupprops

## Contents

## Definition

### As a Sylow subgroup

The term **2-Sylow subgroup of symmetric group** refers to a group that occurs as the 2-Sylow subgroup of a symmetric group on finite set, i.e., a symmetric group on a set of finite size.

For every natural number , there is a corresponding 2-Sylow subgroup of the symmetric group . For an even number and the odd number bigger than it by 1, the 2-Sylow subgroups of the corresponding symmetric groups are isomorphic.

### Explicit description in terms of wreath products

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## Particular cases

symmetric group | order of (= ) | 2-Sylow subgroup | order of 2-Sylow subgroup (= largest power of 2 dividing ) | log to the base 2 of order of 2-Sylow subgroup (= sum of the integer parts of , varying over positive integers) | Information on Sylow subgroup inside symmetric group | |
---|---|---|---|---|---|---|

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

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

2 | cyclic group:Z2 | 2 | cyclic group:Z2 | 2 | 1 | -- |

3 | symmetric group:S3 | 6 | cyclic group:Z2 | 2 | 1 | S2 in S3 |

4 | symmetric group:S4 | 24 | dihedral group:D8 | 8 | 3 | D8 in S4 |

5 | symmetric group:S5 | 120 | dihedral group:D8 | 8 | 3 | D8 in S5 |

6 | symmetric group:S6 | 720 | direct product of D8 and Z2 | 16 | 4 | |

7 | symmetric group:S7 | 5040 | direct product of D8 and Z2 | 16 | 4 | |

8 | symmetric group:S8 | 40320 | wreath product of D8 and Z2 | 128 | 7 | |

9 | symmetric group:S9 | 362880 | wreath product of D8 and Z2 | 128 | 7 |