# Double cover of symmetric group

## 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 $S_n$ only when $n \ge 4$. Further, for each $n$, 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 $H^2(S_n;\mathbb{Z}_2)$ for $n \ge 4$, 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 $2 \cdot S_n^-$, has a presentation with generating set of size $n$ given by $z,s_1,s_2,\dots,s_{n-1}$. The idea is that under the surjective map to $S_n$, $z$ maps to the identity and the relations collapse to the Coxeter presentation of $S_n$. The subgroup $\langle z \rangle$ is the base cyclic group:Z2. The relations (here $e$ denotes the identity element):

Relation Condition on subscripts Number of such relations What it descends to when we quotient to $S_n$
$\! z^2 = e$ no subscripts 1 $e = e$, i.e., a vacuous relation
$\!s_i^2 = z$ $1 \le i \le n - 1$ $n - 1$ $s_i^2 = e$
$\! s_{i+1}s_is_{i+1} = s_is_{i+1}s_iz$ $1 \le i \le n - 2$ $n - 2$ $s_{i+1}s_is_{i+1} = s_is_{i+1}s_i$
$\! s_js_i = s_is_jz$ $1 \le i < j \le n - 1$ and $|i - j| \ge 2$ $(n - 2)(n - 3)/2$ $s_js_i = s_is_j$
Total (--) -- $(n^2 - n + 2)/2$ --

### Presentation for "+" type double cover

This group, denoted $2 \cdot S_n^+$, has a presentation with generating set of size $n$ given by $z,s_1,s_2,\dots,s_{n-1}$. The idea is that under the surjective map to $S_n$, $z$ maps to the identity and the relations collapse to the Coxeter presentation of $S_n$. The subgroup $\langle z \rangle$ is the base cyclic group:Z2. The relations (here $e$ denotes the identity element):

Relation Condition on subscripts Number of such relations What it descends to when we quotient to $S_n$
$\! z^2 = e$ no subscripts 1 $e = e$, i.e., a vacuous relation
$\! s_iz = zs_i$ $1 \le i \le n- 1$ $n - 1$ $s_i = s_i$, i.e., a vacuous relation
$\! s_i^2 = e$ $1 \le i \le n - 1$ $n - 1$ $s_i^2 = e$
$\!s_{i+1}s_is_{i+1} = s_is_{i+1}s_i$ $1 \le i \le n - 2$ $n - 2$ $s_{i+1}s_is_{i+1} = s_is_{i+1}s_i$
$\! s_js_i = s_is_jz$ $1 \le i < j \le n - 1$ and $|i - j| \ge 2$ $(n - 2)(n - 3)/2$ $s_js_i = s_is_j$
Total (--) -- $(n^2 + n)/2$ --

## Particular cases

$n$ order of symmetric group $S_n$ equals $n!$ order of double cover of symmetric group = $2 (n!)$ (twice the preceding column) symmetric group $S_n$ $2 \cdot S_n^-$ (double cover of "-" type) $2 \cdot S_n^+$ (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