# Double cover of symmetric group

## Contents

## Definition

The term **double cover of symmetric group** is used for a stem extension where the base normal subgroup is cyclic group:Z2 and the quotient group is an symmetric group of finite degree.

A double cover exists for the symmetric group only when . Further, for each , there are two possibilities for the double cover, the "+" type and the "-" type.

If we consider the group cohomology of symmetric groups and in particular the second cohomology group for trivial group action for , this second cohomology group is isomorphic to the Klein four-group, and two of the three nontrivial elements of the group correspond to these double covers. The third nontrivial element corresponds to a group extension where the alternating group in the quotient is kept intact and the odd permutations square to the non-identity central element.

In all cases, both double covers are Schur covering groups for the corresponding symmetric group, because the Schur multiplier is precisely cyclic group:Z2.

### Presentation for "-" type double cover

This group, denoted , has a presentation with generating set of size given by . The idea is that under the surjective map to , maps to the identity and the relations collapse to the Coxeter presentation of . The subgroup is the base cyclic group:Z2. The relations (here denotes the identity element):

Relation | Condition on subscripts | Number of such relations | What it descends to when we quotient to |
---|---|---|---|

no subscripts | 1 | , i.e., a vacuous relation | |

and | |||

Total (--) | -- | -- |

### Presentation for "+" type double cover

This group, denoted , has a presentation with generating set of size given by . The idea is that under the surjective map to , maps to the identity and the relations collapse to the Coxeter presentation of . The subgroup is the base cyclic group:Z2. The relations (here denotes the identity element):

Relation | Condition on subscripts | Number of such relations | What it descends to when we quotient to |
---|---|---|---|

no subscripts | 1 | , i.e., a vacuous relation | |

, i.e., a vacuous relation | |||

and | |||

Total (--) | -- | -- |

## Particular cases

order of symmetric group equals | order of double cover of symmetric group = (twice the preceding column) | symmetric group | (double cover of "-" type) | (double cover of "+" type) | Cohomology information | Cohomology group information | |
---|---|---|---|---|---|---|---|

4 | 24 | 48 | symmetric group:S4 | binary octahedral group | general linear group:GL(2,3) | group cohomology of symmetric group:S4 | second cohomology group for trivial group action of S4 on Z2 |

5 | 120 | 240 | symmetric group:S5 | double cover of symmetric group:S5 of minus type | double cover of symmetric group:S5 of plus type | group cohomology of symmetric group:S5 | second cohomology group for trivial group action of S5 on Z2 |