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 $⟨z⟩$ 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
$s_i^2=z$	$1\le i\le n-1$	$n-1$	$s_i^2=e$
$s_{i+1}{s}_i{s}_{i+1}=s_i{s}_{i+1}{s}_{i}z$	$1\le i\le n-2$	$n-2$	$s_{i+1}{s}_i{s}_{i+1}=s_i{s}_{i+1}{s}_i$
$s_j{s}_i=s_i{s}_{j}z$	$1\le i<j\le n-1$ and $|i-j|\ge 2$	$(n-2)(n-3)/2$	$s_j{s}_i=s_i{s}_j$
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 $⟨z⟩$ 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
$s_{i}z=z{s}_i$	$1\le i\le n-1$	$n-1$	$s_i=s_i$, i.e., a vacuous relation
$s_i^2=e$	$1\le i\le n-1$	$n-1$	$s_i^2=e$
$s_{i+1}{s}_i{s}_{i+1}=s_i{s}_{i+1}{s}_i$	$1\le i\le n-2$	$n-2$	$s_{i+1}{s}_i{s}_{i+1}=s_i{s}_{i+1}{s}_i$
$s_j{s}_i=s_i{s}_{j}z$	$1\le i<j\le n-1$ and $|i-j|\ge 2$	$(n-2)(n-3)/2$	$s_j{s}_i=s_i{s}_j$
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