Editing Symmetric group (section)

== Homology ==
{{see also|Alternating group#Group homology}}
The [[group homology]] of S<sub>''n''</sub> is quite regular and stabilizes: the first homology (concretely, the [[abelianization]]) is:

:<math>H_1(\mathrm{S}_n,\mathbf{Z}) = \begin{cases} 0 & n < 2\\
\mathbf{Z}/2 & n \geq 2.\end{cases}</math>

The first homology group is the abelianization, and corresponds to the sign map S<sub>''n''</sub> → S<sub>2</sub> which is the abelianization for ''n'' ≥ 2; for ''n'' < 2 the symmetric group is trivial. This homology is easily computed as follows: S<sub>''n''</sub> is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps {{nowrap|S<sub>''n''</sub> → C<sub>''p''</sub>}} are to S<sub>2</sub> and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps {{nowrap|S<sub>''n''</sub> → S<sub>2</sub> ≅ {±1} }} send an involution to 1 (the trivial map) or to −1 (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of S<sub>''n''</sub>.

The second homology (concretely, the [[Schur multiplier]]) is:
:<math>H_2(\mathrm{S}_n,\mathbf{Z}) = \begin{cases} 0 & n < 4\\
\mathbf{Z}/2 & n \geq 4.\end{cases}</math>
This was computed in {{Harv|Schur|1911}}, and corresponds to the [[covering groups of the alternating and symmetric groups|double cover of the symmetric group]], 2 · S<sub>''n''</sub>.

Note that the [[exceptional object|exceptional]] low-dimensional homology of the alternating group (<math>H_1(\mathrm{A}_3)\cong H_1(\mathrm{A}_4) \cong \mathrm{C}_3,</math> corresponding to non-trivial abelianization, and <math>H_2(\mathrm{A}_6)\cong H_2(\mathrm{A}_7) \cong \mathrm{C}_6,</math> due to the exceptional 3-fold cover) does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map <math>\mathrm{A}_4 \twoheadrightarrow \mathrm{C}_3</math> extends to <math>\mathrm{S}_4 \twoheadrightarrow \mathrm{S}_3,</math> and the triple covers of A<sub>6</sub> and A<sub>7</sub> extend to triple covers of S<sub>6</sub> and S<sub>7</sub> – but these are not ''homological'' – the map <math>\mathrm{S}_4 \twoheadrightarrow \mathrm{S}_3</math> does not change the abelianization of S<sub>4</sub>, and the triple covers do not correspond to homology either.

The homology "stabilizes" in the sense of [[stable homotopy]] theory: there is an inclusion map {{nowrap|S<sub>''n''</sub> → S<sub>''n''+1</sub>}}, and for fixed ''k'', the induced map on homology {{nowrap|''H''<sub>''k''</sub>(S<sub>''n''</sub>) → ''H''<sub>''k''</sub>(S<sub>''n''+1</sub>)}} is an isomorphism for sufficiently high ''n''. This is analogous to the homology of families [[Lie groups]] stabilizing.

The homology of the infinite symmetric group is computed in {{Harv|Nakaoka|1961}}, with the cohomology algebra forming a [[Hopf algebra]].