2-cocycle for a group action

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This term is defined, and makes sense, in the context of a group action on an Abelian group. In particular, it thus makes sense for a linear representation of a group

Definition

Let G be a group acting (on the left) on an abelian group A via a homomorphism of groups \varphi:G \to \operatorname{Aut}(A) where \operatorname{Aut}(A) is the automorphism group of A.

Explicit definition

A 2-cocycle for the action is a function f:G \times G \to A satisfying:

\!\varphi(g_1)(f(g_2,g_3)) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_1,g_2)

If we suppress \varphi and use \cdot for the action, we can rewrite this as:

\!g_1 \cdot f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_1,g_2) or equivalently:

\! g_1 \cdot f(g_2,g_3) - f(g_1g_2,g_3) + f(g_1,g_2g_3) - f(g_1,g_2) = 0

Note that a function f:G \times G \to A (without any conditions) is sometimes termed a 2-cochain for the group action.

Definition as part of the general definition of cocycle

A 2-cocycle for a group action is a special case of a cocycle for a group action, namely n = 2. This, in turn, is the notion of cocycle corresponding to the Hom complex from the bar resolution of G to A as \mathbb{Z}G-modules.

Group structure

The set of 2-cocycles for the action of G on A forms a group under pointwise addition. This group is denoted Z^2_\varphi(G,A) where \varphi is the action. If the action is understood from the context, it can simply be denoted as Z^2(G,A).

As a group of homomorphisms

For any group G, we can construct a \mathbb{Z}G-module K such that for any abelian group A, the group of 2-cocycles f:G \times G \to A can be identified with the group of \mathbb{Z}G-module maps from K to A.

Further information: group of 2-cocycles is naturally identified with group of homomorphisms from a particular module

Particular cases and variations

Case or variation Condition for 2-coycle Further information
G acts trivially on A (i.e., every element of G fixes every element of A) \! f(g_2,g_3) + f(g_1,g_2g_3) = f(g_1g_2,g_3) + f(g_1,g_2) 2-cocycle for trivial group action
G is abelian with additive notation for its group operation, and it acts trivially on A \! f(g_2,g_3) + f(g_1,g_2 + g_3) = f(g_1 + g_2,g_3) + f(g_1,g_2)

Examples

Extreme examples

  • If A is the trivial group, the group of 2-cocycles is the trivial group, with the only 2-cocycle being the map that sends every pair of elements of G to the zero element of A.
  • If G is the trivial group, the group of 2-cocycles is isomorphic to the group A, with each 2-cocycle being identified with its image value in A.

Other examples

Acting group G Group A acted upon Action Group of 2-cocycles
trivial group any group A trivial action isomorphic to A, where a 2-cocycle is identified with its image point
any group G trivial group trivial action trivial group -- the unique 2-cocycle that sends everything to the zero element
cyclic group:Z2 cyclic group:Z2 trivial action  ?

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Capture of difference Intermediate notions
2-coboundary for a group action f(g,h) := g \cdot \alpha(h) - \alpha(gh) + \alpha(g) for some function \alpha:G \to A 2-coboundary implies 2-cocycle 2-cocycle not implies 2-coboundary The quotient of the group of 2-cocycles by the group of 2-coboundaries is termed the second cohomology group |FULL LIST, MORE INFO
symmetric 2-cocycle for a group action f is a 2-cocycle and is a symmetric function of both its inputs
normalized 2-cocycle for a group action f is a 2-cocycle and if either of the inputs to f is the identity element of f, the output of f is zero

Importance

Extension involving an abelian normal subgroup

Let E be a group with an abelian normal subgroup isomorphic to (and explicitly identified with) A, and a quotient isomorphic to (and explicitly identified with) G, such that the induced action of the quotient G on A (in the sense of action by conjugation, see quotient group acts on abelian normal subgroup) is as described. Let S be a system of coset representatives for G in E with s: G \to S being the representation map. Then, define f: G \times G \to A such that

\! s(gh) = f(g,h)s(g)s(h)

In other words, f measures the extent to which the collection of coset representatives fails to be closed under multiplication.

Such an f is a 2-cocycle.

Note that for a particular choice of E, all the 2-cocycles obtained by different choices of S will form a single coset of the coboundary group, that is, any two such cocycles will differ by a coboundary. Thus, in particular, we can intrinsically associate, to every extension E with abelian normal subgroup A and quotient G, an element of the second cohomology group.

Further information: second cohomology group