1-coboundary 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


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, i.e., there exists a homomorphism of groups \varphi:G \to \operatorname{Aut}(A) where \operatorname{Aut}(A) is the automorphism group of A.

A 1-coboundary, also called a principal crossed homomorphism, for this group action is a function f:G \to A such that there exists a a \in A such that for all g \in G:

f(g) = \varphi(g)(a) - a

If we denote the action by \cdot, this can be rewritten as:

f(g) = g \cdot a - a

The 1-coboundaries form an Abelian group under pointwise addition of functions.

Importance

Suppose E is a group having A as a normal subgroup with G as the quotient group. Then, 1-coboundaries in E correspond to inner automorphisms by elements of A which are in the stability group of the ascending series 1 \triangleleft A \triangleleft E.

In fact, the a for the inner automorphism and the coboundary is the same.

The 1-coboundary group is thus a quotient of the group A itself, by the subgroup of A comprising G-invariant elements. Two particular cases:

  • The action of G on A is faithful: In this case the 1-coboundary group is isomorphic to A
  • The action of G on A is trivial: In this case the 1-coboundary group is trivial

Further information: 1-cocycle for a group action