Second cohomology group

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This article gives a basic definition in the following area: group cohomology
View other basic definitions in group cohomology |View terms related to group cohomology |View facts related to group cohomology

Definition

Let G be a group acting on an abelian group A, via an action \varphi:G \to \operatorname{Aut}(A). Equivalently, A is a module over the (possibly non-commutative) unital group ring \mathbb{Z}G of G over the ring of integers.

Definition in cohomology terms

The second cohomology group H^2_\varphi(G,A) (also denoted H^2_\varphi(G;A)) is an abelian group defined in the following equivalent ways.

When \varphi is understood from context, the subscript {}_\varphi may be omitted in the notation for the cohomology group, as well as the notation for the groups of 2-cocycles and 2-coboundaries.

No. Shorthand Detailed description of H^2_\varphi(G,A), the second cohomology group
1 Explicit, using the bar resolution H^2_\varphi(G,A), is defined as the quotient Z^2_\varphi(G,A)/B^2_\varphi(G,A) where Z^2_\varphi(G,A) is the group of 2-cocycles for the action and B^2_\varphi(G,A) is the group of 2-coboundaries.
1' Explicit, using the normalized bar resolution Same as definition (1), but we use normalized cocycles and normalized coboundaries instead of arbitrary cocycles and coboundaries.
2 Complex based on arbitrary resolution Let \mathcal{F} be a projective resolution for \mathbb{Z} as a \mathbb{Z}G-module with the trivial action. Let \mathcal{C} be the complex \operatorname{Hom}_{\mathbb{Z}G}(\mathcal{F},A). The cohomology group H^2_\varphi(G,A) is defined as the second cohomology group for this complex.
3 Complex based on arbitrary injective resolution (works if category of \mathbb{Z}G-modules has enough injectives!) Let \mathcal{I} be an injective resolution for A as a \mathbb{Z}G-module with the specified action \varphi. Let \mathcal{D} be the complex \operatorname{Hom}_{\mathbb{Z}G}(\mathbb{Z},\mathcal{I}) where \mathbb{Z} has the structure of a trivial action \mathbb{Z}G-module. The cohomology group H^2_\varphi(G,A) is defined as the second cohomology group for this complex.
4 As an \operatorname{Ext}^2 functor \operatorname{Ext}^2_{\mathbb{Z}G}(\mathbb{Z},A) where \mathbb{Z} is a trivial \mathbb{Z}G-module and A has the module structure specified by \varphi.
5 As a right derived functor H^2_\varphi(G,A) = R^2(-^G)(A), i.e., it is the second right derived functor of the invariants functor for G (denoted -^G) evaluated at A. The invariants functor sends a \mathbb{Z}G-module to its submodule of elements fixed by all elements of G.

All these definitions have natural analogues for the n^{th} cohomology group H^n_\varphi(G,A) for all n \ge 0. For more, see cohomology group.

Definition in terms of group extensions

There is an alternative definition of H^2_\varphi(G,A) that is specific to 2 and has no easy analogue for other H^n_\varphi(G,A). This is in terms of group extensions.

H^2_\varphi(G,A) can also be identified with the set of congruence classes of group extensions with normal subgroup isomorphic to A and quotient group isomorphic to G where the induced action of the quotient is the specified action \varphi. By a group extension, we mean a group E having A as a normal subgroup and G as a quotient group. Two extensions E_1 and E_2 are congruent if there is an isomorphism of E_1 to E_2 which is identity on A and induces the identity map on G as a quotient.

Equivalence of the definitions

Further information: Equivalence of definitions of second cohomology group

Particular cases

A very special case where a lot of additional things of interest happen is that where the action is trivial. See second cohomology group for trivial group action. In particular, in the case of a trivial action, the second cohomology group coincides with the second cohomology group of the classifying space of G with coefficients in A.

Functoriality and automorphisms

Covariance in the second group

Suppose G is a group and A_1,A_2 are abelian groups. Suppose \varphi_1:G \to \operatorname{Aut}(A_1), \varphi_2:G \to \operatorname{Aut}(A_2), and \alpha:A_1 \to A_2 are group homomorphisms such that \alpha \circ \varphi_1(g) = \varphi_2(g) \circ \alpha for all g \in G.

In other words, \alpha is a homomorphism from the G-module A_1 with action \varphi_1 to the G-module A_2 with action \varphi_2.

Then, we get an induced homomorphism between the second cohomology groups:

H^2(\alpha): H^2_{\varphi_1}(G,A_1) \to H^2_{\varphi_2}(G,A_2)

This association is functorial, i.e., it gives a (covariant) functor from the category of \mathbb{Z}G-modules (i.e., abelian groups with G acting on them) to the category of abelian groups.

Contravariance in the first group

Further information: restriction functor on cohomology, inflation functor on cohomology

Suppose G_1,G_2 are groups and A is an abelian group. Suppose \varphi_1:G_1 \to \operatorname{Aut}(A), \varphi_2: G_2 \to \operatorname{Aut}(A), and \alpha: G_1 \to G_2 are homomorphisms such that \varphi_2 \circ \alpha = \varphi_1, i.e., the G_1-action and G_2-action on A are compatible. Then, we get an induced homomorphism between the second cohomology groups:

\operatorname{res}^{G_2}_{G_1}: H^2_{\varphi_2}(G_2,A) \to H^2_{\varphi_1}(G_1,A)

Note that the direction of this homomorphism is reverse to the direction of the original homomorphism. The association gives a contravariant functor. The functor in general is termed the restriction functor.

Automorphism group actions

  • Due to the covariance in the second argument, there is a natural action on H^2_\varphi(G,A) of the group C_{\operatorname{Aut}(A)}(G) = C_{\operatorname{Aut}(A)}(\varphi(G)), i.e., the subgroup of the automorphism group of A comprising those automorphisms that commute with the action of G.
  • Due to the contravariance in the first argument, there is a natural action on H^2_\varphi(G,A) of the subgroup of \operatorname{Aut}(G) that send every coset of the subgroup C_G(A) to itself (or equivalently, induce the identity map on G/C_G(A). Here, C_G(A) is a normal subgroup of G defined as the kernel of \varphi.

Examples

Here, we use the notation with G a group acting on an abelian group A via a group action \varphi.

Extreme examples

  • If G is a trivial group, then the second cohomology group H^2_\varphi(G,A) is also a trivial group.
  • If A is a trivial group, then the second cohomology group H^2_\varphi(G,A) is also a trivial group.

Other examples

Acting group G Group A acted upon Action Second cohomology group Groups obtained as extensions More information
cyclic group:Z2 cyclic group:Z2 trivial action cyclic group:Z2 Klein four-group and cyclic group:Z4 second cohomology group for trivial group action of Z2 on Z2
cyclic group:Z2 cyclic group:Z4 trivial action cyclic group:Z2 direct product of Z4 and Z2 and cyclic group:Z8 second cohomology group for trivial group action of Z2 on Z4
cyclic group:Z2 cyclic group:Z4 non-identity element acts by inverse map cyclic group:Z2 dihedral group:D8 and quaternion group second cohomology group for nontrivial group action of Z2 on Z4
cyclic group:Z2 Klein four-group trivial action Klein four-group elementary abelian group:E8, direct product of Z4 and Z2 (occurs in three ways) second cohomology group for trivial group action of Z2 on V4
cyclic group:Z2 Klein four-group non-identity element acts by exchanging coordinates trivial group dihedral group:D8 second cohomology group for nontrivial group action of Z2 on V4
cyclic group:Z4 cyclic group:Z2 trivial action cyclic group:Z2 direct product of Z4 and Z2 and cyclic group:Z8 second cohomology group for trivial group action of Z4 on Z2
Klein four-group cyclic group:Z2 trivial action elementary abelian group:E8 elementary abelian group:E8, direct product of Z4 and Z2 (3 times), dihedral group:D8 (3 times), quaternion group second cohomology group for trivial group action of V4 on Z2
cyclic group:Z2 cyclic group:Z8 trivial action cyclic group:Z2 direct product of Z8 and Z2 and cyclic group:Z16 second cohomology group for trivial group action of Z2 on Z8
cyclic group:Z2 cyclic group:Z8 non-identity element acts by inverse map cyclic group:Z2 dihedral group:D16 and generalized quaternion group:Q16 second cohomology group for inverse map action of Z2 on Z8
cyclic group:Z2 cyclic group:Z8 non-identity element acts by cube map trivial group semidihedral group:SD16  ?
cyclic group:Z2 cyclic group:Z8 non-identity element acts by the fifth power map trivial group M16  ?
cyclic group:Z4 cyclic group:Z4 trivial action cyclic group:Z4 direct product of Z4 and Z4, cyclic group:Z16 (occurs in two ways), direct product of Z8 and Z2 second cohomology group for trivial group action of Z4 on Z4
Klein four-group cyclic group:Z4 trivial action elementary abelian group:E8 direct product of Z4 and V4, direct product of Z8 and Z2, central product of D8 and Z4, M16 second cohomology group for trivial group action of V4 on Z4
Klein four-group cyclic group:Z4 one coordinate acts by inverse map, other coordinate acts trivially  ? direct product of D8 and Z2, direct product of Q8 and Z2, central product of D8 and Z4, dihedral group:D16, semidihedral group:SD16, generalized quaternion group:Q16
Klein four-group Klein four-group trivial action elementary abelian group:E64 elementary abelian group:E16, direct product of Z4 and Z4, direct product of Z4 and V4, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, direct product of D8 and Z2, direct product of Q8 and Z2 second cohomology group for trivial group action of V4 on V4
cyclic group:Z8 cyclic group:Z2 trivial action cyclic group:Z2 direct product of Z8 and Z2 and cyclic group:Z16 second cohomology group for trivial group action of Z8 on Z2
direct product of Z4 and Z2 cyclic group:Z2 trivial action elementary abelian group:E8 direct product of Z4 and V4, direct product of Z8 and Z2, direct product of Z4 and Z4, SmallGroup(16,3), M16, nontrivial semidirect product of Z4 and Z4 second cohomology group for trivial group action of direct product of Z4 and Z2 on Z2
dihedral group:D8 cyclic group:Z2 trivial action elementary abelian group:E8 direct product of D8 and Z2, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, dihedral group:D16, generalized quaternion group:Q16, semidihedral group:SD16 second cohomology group for trivial group action of D8 on Z2
quaternion group cyclic group:Z2 trivial action Klein four-group direct product of Q8 and Z2, nontrivial semidirect product of Z4 and Z4 second cohomology group for trivial group action of Q8 on Z2
elementary abelian group:E8 cyclic group:Z2 trivial action elementary abelian group:E64 elementary abelian group:E16, direct product of Z4 and V4, direct product of Q8 and Z2, central product of D8 and Z4, direct product of D8 and Z2 second cohomology group for trivial group action of E8 on Z2
Klein four-group cyclic group:Z8 trivial action elementary abelian group:E8 direct product of Z8 and V4, central product of D8 and Z8, direct product of Z16 and Z2, M32 second cohomology group for trivial group action of V4 on Z8