Group cohomology of symmetric groups

This article gives specific information, namely, group cohomology, about a family of groups, namely: symmetric group.
View group cohomology of group families | View other specific information about symmetric group

Particular cases

$n$	$n!$ (order of symmetric group)	Symmetric group of degree $n$	Group cohomology page
1	1	trivial group	group cohomology of trivial group
2	2	cyclic group:Z2	group cohomology of cyclic group:Z2
3	6	symmetric group:S3	group cohomology of symmetric group:S3
4	24	symmetric group:S4	group cohomology of symmetric group:S4
5	120	symmetric group:S5	group cohomology of symmetric group:S5
6	720	symmetric group:S6	group cohomology of symmetric group:S6

Classifying space and corresponding chain complex

The homology and cohomology groups of the symmetric group $S_{n}$ are the same as the respective homology and cohomology groups of the configuration space of $n$ unordered points in a countable-dimensional real projective space. For more on the topological perspective, see configuration space of unordered points of a countable-dimensional real projective space on the Topospaces wiki.

Homology groups for trivial group action

FACTS TO CHECK AGAINST (homology group for trivial group action):
First homology group: first homology group for trivial group action equals tensor product with abelianization
Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier
General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

Over the integers

Here, "0" for a group is shorthand for the trivial group. $\mathbb {Z} _{m}$ is shorthand for the finite cyclic group $\mathbb {Z} /m\mathbb {Z}$.

The homology groups eventually stabilize in the following sense: for any fixed $q$, there exists a large enough $n$ (explicit expression for $n$ in terms of $q$ -- around double?) such that $H_{q}(S_{m};\mathbb {Z} )\cong H_{q}(S_{n};\mathbb {Z} )$ for all $m\geq n$. The corresponding stable value of homology group is termed the stable homology group of degree $q$ for the symmetric groups.

$n$	$n!$	symmetric group $S_{n}$ of degree $n$	if a finite group with periodic cohomology, period of sequence of homology groups for positive degrees	$H_{1}(S_{n};\mathbb {Z} )$	$H_{2}(S_{n};\mathbb {Z} )$	$H_{3}(S_{n};\mathbb {Z} )$	$H_{4}(S_{n};\mathbb {Z} )$	$H_{5}(S_{n};\mathbb {Z} )$	$H_{6}(S_{n};\mathbb {Z} )$	$H_{7}(S_{n};\mathbb {Z} )$	$H_{8}(S_{n};\mathbb {Z} )$	$H_{9}(S_{n};\mathbb {Z} )$	$H_{10}(S_{n};\mathbb {Z} )$
0	1	trivial group	1	0	0	0	0	0	0	0	0	0	0
1	1	trivial group	1	0	0	0	0	0	0	0	0	0	0
2	2	cyclic group:Z2	2	$\mathbb {Z} _{2}$	0	$\mathbb {Z} _{2}$	0	$\mathbb {Z} _{2}$	0	$\mathbb {Z} _{2}$	0	$\mathbb {Z} _{2}$	0
3	6	symmetric group:S3	4	$\mathbb {Z} _{2}$	0	$\mathbb {Z} _{6}$	0	$\mathbb {Z} _{2}$	0	$\mathbb {Z} _{6}$	0	$\mathbb {Z} _{2}$	0
4	24	symmetric group:S4	--	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{12}\oplus \mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$(\mathbb {Z} _{2})^{3}$	$(\mathbb {Z} _{2})^{2}$	$\mathbb {Z} _{12}\oplus (\mathbb {Z} _{2})^{2}$	$(\mathbb {Z} _{2})^{3}$	$(\mathbb {Z} _{2})^{4}$	$(\mathbb {Z} _{2})^{3}$
5	120	symmetric group:S5	--	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{12}\oplus \mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$(\mathbb {Z} _{2})^{3}$	$(\mathbb {Z} _{2})^{2}$	$\mathbb {Z} _{60}\oplus (\mathbb {Z} _{2})^{2}$	$(\mathbb {Z} _{2})^{3}$	$(\mathbb {Z} _{2})^{4}$	$(\mathbb {Z} _{2})^{3}$
6	720	symmetric group:S6	--	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{12}\oplus \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}$	$\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}$	$(\mathbb {Z} _{2})^{5}$	$(\mathbb {Z} _{2})^{5}$	$\mathbb {Z} _{60}\oplus \mathbb {Z} _{6}\oplus (\mathbb {Z} _{2})^{5}$	$(\mathbb {Z} _{2})^{8}$
7	5040	symmetric group:S7	--	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{12}\oplus \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}$	$\mathbb {Z} _{2}\oplus \mathbb {Z} _{2}$	$(\mathbb {Z} _{2})^{5}$	$(\mathbb {Z} _{2})^{5}$
Stable	--	--	--	$\mathbb {Z} _{2}$	$\mathbb {Z} _{2}$	$\mathbb {Z} _{12}\oplus \mathbb {Z} _{2}\oplus \mathbb {Z} _{2}$	?	?	?	?	?	?	?