Second cohomology group

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Definition

Let $G$ be a group acting on an abelian group $A$, via an action $\phi :G\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$. 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 $H_{\phi }^2(G,A)$ (also denoted $H_{\phi }^2(G;A)$) is an abelian group defined in the following equivalent ways.

When $\phi$ is understood from context, the subscript ${}_{\phi }$ 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 $H_{\phi }^2(G,A)$, the second cohomology group
1	Explicit, using the bar resolution	$H_{\phi }^2(G,A)$, is defined as the quotient $Z_{\phi }^2(G,A)/B_{\phi }^2(G,A)$ where $Z_{\phi }^2(G,A)$ is the group of 2-cocycles for the action and $B_{\phi }^2(G,A)$ 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 ${\mathrm{H}}_{\mathrm{o}}(\mathcal{F},A)$. The cohomology group $H_{\phi }^2(G,A)$ 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 $\phi$. Let $\mathcal{D}$ be the complex ${\mathrm{H}}_{\mathrm{o}}(\mathbb{Z},\mathcal{I})$ where $\mathbb{Z}$ has the structure of a trivial action $\mathbb{Z}G$-module. The cohomology group $H_{\phi }^2(G,A)$ is defined as the second cohomology group for this complex.
4	As an $\stackrel{\mathrm{x}}{\mathrm{E}}$ functor	${\mathrm{E}}_{\mathrm{x}}^{\mathrm{t}}(\mathbb{Z},A)$ where $\mathbb{Z}$ is a trivial $\mathbb{Z}G$-module and $A$ has the module structure specified by $\phi$.
5	As a right derived functor	$H_{\phi }^2(G,A)=R^2({-}^G)(A)$, 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 $H_{\phi }^n(G,A)$ for all $n\ge 0$. For more, see cohomology group.

Definition in terms of group extensions

There is an alternative definition of $H_{\phi }^2(G,A)$ that is specific to 2 and has no easy analogue for other $H_{\phi }^n(G,A)$. This is in terms of group extensions.

$H_{\phi }^2(G,A)$ 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 $\phi$. 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 ${\phi }_1:G\to \mathrm{A}\mathrm{u}\mathrm{t}(A_1)$, ${\phi }_2:G\to \mathrm{A}\mathrm{u}\mathrm{t}(A_2)$, and $\alpha :A_1\to A_2$ are group homomorphisms such that $\alpha \circ {\phi }_1(g)={\phi }_2(g)\circ \alpha$ for all $g\in G$.

In other words, $\alpha$ is a homomorphism from the $G$-module $A_1$ with action ${\phi }_1$ to the $G$-module $A_2$ with action ${\phi }_2$.

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

$H^2(\alpha ):H_{{\phi }_1}^2(G,A_1)\to H_{{\phi }_2}^2(G,A_2)$

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 ${\phi }_1:G_1\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$, ${\phi }_2:G_2\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$, and $\alpha :G_1\to G_2$ are homomorphisms such that ${\phi }_2\circ \alpha ={\phi }_1$, 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:

${\mathrm{r}}_{\mathrm{e}}^{\mathrm{s}}:H_{{\phi }_2}^2(G_2,A)\to H_{{\phi }_1}^2(G_1,A)$

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 $H_{\phi }^2(G,A)$ of the group $C_{\mathrm{A}\mathrm{u}\mathrm{t}(A)}(G)=C_{\mathrm{A}\mathrm{u}\mathrm{t}(A)}(\phi (G))$, 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 $H_{\phi }^2(G,A)$ of the subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(G)$ 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 $\phi$.

Examples

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

Extreme examples

• If $G$ is a trivial group, then the second cohomology group $H_{\phi }^2(G,A)$ is also a trivial group.
• If $A$ is a trivial group, then the second cohomology group $H_{\phi }^2(G,A)$ 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