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Vipul at 22:57, 3 October 2007

2007-10-03T22:57:36Z

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{{basicdef in|group cohomology}}

{{group action on Abelian group}}

==Definition==

Let <math>G</math> be a [[group]] acting on an [[Abelian group]] <math>A</math>. A '''2-coboundary''' for the action of <math>G</math> on <math>A</math>, is a function <math>f: G \times G \to A</math> such that there exists a function <math>\phi:G \to A</math> such that:

<math>f = (g,h) \mapsto g.\phi(h) - \phi(gh) + \phi(g)</math>

==Importance==

Suppose we want to classify all groups <math>E</math> which arise as the [[group extension]] with normal subgroup <math>A</math> and quotient <math>G</math>. One approach to describing such a group <math>E</math> is to define a collection <math>S</math> of coset representatives for <math>A</math> in <math>E</math>. This can be viewed as a map from <math>G</math> to <math>E</math>. Call the coset representative for <math>g</math> <math>b_g</math>. Define <math>f(g,h)</math> as the element of <math>A</math> given by <math>b_{gh}^{-1}b_gb_h</math>.

It turns out that if we pick two different collections of coset representatives and let <math>f_1</math> and <math>f_2</math> be the functions corresponding to them, then <math>f_1 - f_2</math> is a 2-coboundary.

It's also true that each of <math>f_1</math> and <math>f_2</math> needs to be a [[2-cocycle for a group action|2-cocycle]], and thus the collection of possible extensions of <math>G</math> by <math>A</math> is classified by the [[second cohomology group]] for the action of <math>G</math> on <math>A</math>.

{{further|[[second cohomology group]]}}

2-coboundary for a group action - Revision history

Vipul: 1 revision

Vipul at 22:57, 3 October 2007