Double cover of symmetric group

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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