Group cohomology of symmetric groups

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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 \ge 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  ?  ?  ?  ?  ?  ?  ?