Double cover of symmetric group: Difference between revisions

Latest revision as of 22:07, 16 February 2013

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 $n\geq 4$ . 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 $n\geq 4$ , 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
$\!s_{i}^{2}=z$	$1\leq i\leq n-1$	$n-1$	$s_{i}^{2}=e$
$\!s_{i+1}s_{i}s_{i+1}=s_{i}s_{i+1}s_{i}z$	$1\leq i\leq n-2$	$n-2$	$s_{i+1}s_{i}s_{i+1}=s_{i}s_{i+1}s_{i}$
$\!s_{j}s_{i}=s_{i}s_{j}z$	$1\leq i<j\leq n-1$ and $\|i-j\|\geq 2$	$(n-2)(n-3)/2$	$s_{j}s_{i}=s_{i}s_{j}$
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 $\langle z\rangle$ 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
$\!s_{i}z=zs_{i}$	$1\leq i\leq n-1$	$n-1$	$s_{i}=s_{i}$ , i.e., a vacuous relation
$\!s_{i}^{2}=e$	$1\leq i\leq n-1$	$n-1$	$s_{i}^{2}=e$
$\!s_{i+1}s_{i}s_{i+1}=s_{i}s_{i+1}s_{i}$	$1\leq i\leq n-2$	$n-2$	$s_{i+1}s_{i}s_{i+1}=s_{i}s_{i+1}s_{i}$
$\!s_{j}s_{i}=s_{i}s_{j}z$	$1\leq i<j\leq n-1$ and $\|i-j\|\geq 2$	$(n-2)(n-3)/2$	$s_{j}s_{i}=s_{i}s_{j}$
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

@@ Line 1: / Line 1: @@
+[[importance rank::3| ]]
 ==Definition==
@@ Line 16: / Line 17: @@
 ! Relation !! Condition on subscripts !! Number of such relations !! What it descends to when we quotient to <math>S_n</math>
 |-
-| <math>z^2 = e</math> || no subscripts || 1 || <math>e = e</math>, i.e., a vacuous relation
+| <math>\! z^2 = e</math> || no subscripts || 1 || <math>e = e</math>, i.e., a vacuous relation
 |-
-| <math>s_i^2 = z</math> || <math>1 \le i \le n - 1</math> || <math>n - 1</math> || <math>s_i^2 = e</math>
+| <math>\!s_i^2 = z</math> || <math>1 \le i \le n - 1</math> || <math>n - 1</math> || <math>s_i^2 = e</math>
 |-
-| <math>s_{i+1}s_is_{i+1} = s_is_{i+1}s_iz</math> || <math>1 \le i \le n - 2</math> || <math>n - 2</math> || <math>s_{i+1}s_is_{i+1} = s_is_{i+1}s_i</math>
+| <math>\! s_{i+1}s_is_{i+1} = s_is_{i+1}s_iz</math> || <math>1 \le i \le n - 2</math> || <math>n - 2</math> || <math>s_{i+1}s_is_{i+1} = s_is_{i+1}s_i</math>
 |-
-| <math>s_js_i = s_is_jz</math> || <math>1 \le i < j \le n - 1</math> and <math>|i - j| \ge 2</math> || <math>(n - 2)(n - 3)/2</math> || <math>s_js_i = s_is_j</math>
+| <math>\! s_js_i = s_is_jz</math> || <math>1 \le i < j \le n - 1</math> and <math>|i - j| \ge 2</math> || <math>(n - 2)(n - 3)/2</math> || <math>s_js_i = s_is_j</math>
 |-
 | Total (--) || -- || <math>(n^2 - n + 2)/2</math> || --
@@ Line 29: / Line 30: @@
 ===Presentation for "+" type double cover===
-This group, denoted <math>2 \cdot S_n^-</math>, has a presentation with generating set of size <math>n</math> given by <math>z,s_1,s_2,\dots,s_{n-1}</math>. The idea is that under the surjective map to <math>S_n</math>, <math>z</math> maps to the identity and the relations collapse to the Coxeter presentation of <math>S_n</math>. The subgroup <math>\langle z \rangle</math> is the base [[cyclic group:Z2]]. The relations (here <math>e</math> denotes the identity element):
+This group, denoted <math>2 \cdot S_n^+</math>, has a presentation with generating set of size <math>n</math> given by <math>z,s_1,s_2,\dots,s_{n-1}</math>. The idea is that under the surjective map to <math>S_n</math>, <math>z</math> maps to the identity and the relations collapse to the Coxeter presentation of <math>S_n</math>. The subgroup <math>\langle z \rangle</math> is the base [[cyclic group:Z2]]. The relations (here <math>e</math> denotes the identity element):
 {| class="sortable" border="1"
 ! Relation !! Condition on subscripts !! Number of such relations !! What it descends to when we quotient to <math>S_n</math>
 |-
-| <math>z^2 = e</math> || no subscripts || 1 || <math>e = e</math>, i.e., a vacuous relation
+| <math>\! z^2 = e</math> || no subscripts || 1 || <math>e = e</math>, i.e., a vacuous relation
 |-
-| <math>s_i^2 = e</math> || <math>1 \le i \le n - 1</math> || <math>n - 1</math> || <math>s_i^2 = e</math>
+| <math>\! s_iz = zs_i</math> || <math>1 \le i \le n- 1</math> || <math>n - 1</math> || <math>s_i = s_i</math>, i.e., a vacuous relation
 |-
-| <math>s_{i+1}s_is_{i+1} = s_is_{i+1}s_i</math> || <math>1 \le i \le n - 2</math> || <math>n - 2</math> || <math>s_{i+1}s_is_{i+1} = s_is_{i+1}s_i</math>
+| <math>\! s_i^2 = e</math> || <math>1 \le i \le n - 1</math> || <math>n - 1</math> || <math>s_i^2 = e</math>
 |-
-| <math>s_js_i = s_is_jz</math> || <math>1 \le i < j \le n - 1</math> and <math>|i - j| \ge 2</math> || <math>(n - 2)(n - 3)/2</math> || <math>s_js_i = s_is_j</math>
+| <math>\!s_{i+1}s_is_{i+1} = s_is_{i+1}s_i</math> || <math>1 \le i \le n - 2</math> || <math>n - 2</math> || <math>s_{i+1}s_is_{i+1} = s_is_{i+1}s_i</math>
 |-
-| Total (--) || -- || <math>(n^2 - n + 2)/2</math> || --
+| <math>\! s_js_i = s_is_jz</math> || <math>1 \le i < j \le n - 1</math> and <math>|i - j| \ge 2</math> || <math>(n - 2)(n - 3)/2</math> || <math>s_js_i = s_is_j</math>
+|-
+| Total (--) || -- || <math>(n^2 + n)/2</math> || --
 |}
@@ Line 48: / Line 51: @@
 {| class="sortable" border="1"
-! <math>n</math> !! order of symmetric group <math>S_n</math> !! order of double cover of symmetric group = <math>2 (n!)</math> !! symmetric group <math>S_n</math> !! <math>2 \cdot S_n^-</math> (double cover of "-" type) !! <math>2 \cdot S_n^+</math> (double cover of "+" type) !! Cohomology information !! Cohomology group information
+! <math>n</math> !! order of symmetric group <math>S_n</math> equals <math>n!</math> !! order of double cover of symmetric group = <math>2 (n!)</math> (twice the preceding column) !! symmetric group <math>S_n</math> !! <math>2 \cdot S_n^-</math> (double cover of "-" type) !! <math>2 \cdot S_n^+</math> (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]] || ? || ? || [[group cohomology of symmetric group:S5]] || [[second cohomology group for trivial group action of S5 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]]
 |}