Restriction then corestriction is multiplication by the index
Restriction sends cohomology of G to cohomology of a subgroup H; corestriction, or transfer, goes back the other way when the index is finite. Their composite is multiplication by [G : H]. That single identity forces cohomology of a finite group to be annihilated by its order, and it makes cohomology detectable on Sylow subgroups — which is how modular computations are actually organised.
Learning objectives
- Define restriction and corestriction.
- State and use the index formula.
- Deduce that |G| annihilates positive-degree cohomology.
- Explain Sylow detection and stable elements.
Section 01The two maps
Induced by the inclusion ℤ[H] ⊆ ℤ[G]. Defined for any subgroup, of any index.
Defined only for finite index, by summing over coset representatives. Also called the transfer.
Take H = 1. Then Hn(1, A) = 0 for n ≥ 1, so the composite is zero; but it is also multiplication by |G|. Hence |G| annihilates Hn(G, A) for all n ≥ 1. Cohomology of a finite group is torsion in positive degrees, with exponent dividing |G|.
Section 02Sylow detection
- Fix a prime p and let Gp be a Sylow p-subgroup.
- The index [G : Gp] is prime to p, so cor ∘ res is multiplication by a unit on the p-primary part.
- Hence restriction is injective on the p-primary component of Hn(G, A). Nothing p-primary is lost by passing to the Sylow subgroup.
- The image is characterised as the stable elements — those whose restrictions to intersections of conjugates agree.
- So Hn(G, A)(p) is the stable subring of Hn(Gp, A).
Because Sylow detection means the cohomology of every finite group is assembled from the cohomology of p-groups. Computing the cohomology of p-groups is therefore the central problem, and it remains hard.
Section 03Further properties
| Property | Statement |
|---|---|
| Transitivity | res and cor compose correctly through a chain of subgroups |
| Projection formula | cor(x · res y) = cor(x) · y — corestriction is a module map over H*(G) |
| Double coset formula | res ∘ cor decomposes as a sum over double cosets — the source of fusion conditions |
| Normal subgroups | The image of restriction to a normal subgroup lies in the invariants under the quotient action |
| Homology version | Both maps exist with the arrows reversed, and the index formula holds in the same form |
The image lands in the Q-invariants of the subgroup's cohomology, but need not fill them. The obstruction is the transgression in the five-term sequence, and assuming surjectivity is a common error.
ReferenceFrequently asked questions
Does corestriction exist for infinite index?
No — the construction sums over cosets, which requires finitely many. For infinite index only restriction is available, which is why the annihilation results are specific to finite groups.
What are stable elements?
Classes on a Sylow subgroup whose restrictions to intersections with conjugate subgroups agree. They are exactly the classes in the image of restriction from the whole group, and the condition encodes the fusion of the group.
Why is cohomology of a finite group torsion?
Because the index formula with the trivial subgroup shows |G| kills everything in positive degree. Consequently, with coefficients in a ℚ-vector space or any module where |G| is invertible, all higher cohomology vanishes.
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