Page 109 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
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mož Moravec: Some Topics in the Theory of Finite Groups 97
Choose a transversal function τ: G → E , i.e., τε = 1G . Then the above action can be
rewritten as
(a g )µ = g −τa µ g τ.
Let x , y ∈ G . As x τy τ and (x y )τ belong to the same coset of ker ε = im µ in E , we may
write
x τy τ = (x y )τ((x , y )φ)µ
for some (x , y )φ ∈ A. Thus we get a function φ : G × G → A defined by
((x , y )φ)µ = (x y )−τx τy τ.
From the associative law x τ(y τz τ) = (x τy τ)z τ we get that φ satisfies the identity
(x , y z )φ + (y , z )φ = (x y , z )φ + (x , y )φ · z .
A function φ : G × G → A satisfiying this functional equation is called a factor set (or a 2-
cocycle). Note that we can assume without loss of generality that 1τ = 1, therefore we can
always assume that (1, x )φ = (x , 1)φ = 0 for all x ∈ G . The set Z 2(G , A) of all 2-cocycles
in G with coefficients in the G -module A has the structure of an abelian group with the
operation
(x , y )(φ1 + φ2) = (x , y )φ1 + (x , y )φ2.
Example 3.4.5 In the situation above, what happens if (x , y )φ = 0 for all x , y ∈ G ? In this
case, the transversal map τ: G → E is a homomorphism. It is easy to see that the image of
τ is then a complement of im µ ∼= A in E , therefore E =∼ G χ A.
How does the choice of τ affect φ? Let τ be another transversal function for given
extension. Then we get another factor set φ , i.e., x τ y τ = (x y )τ ((x , y )φ )µ. As x τ and x τ
belong to the same coset of ker ε = im µ, we can write
x τ = x τ((x )ψ)µ
for some (x )ψ ∈ A. We get
(x , y )φ = (x , y )φ + (x y )ψ − (x )ψ · y − (y )ψ.
Define ψ∗ : G × G → A by
(x , y )ψ∗ = (y )ψ − (x y )ψ + (x )ψ · y ,
so that φ = φ+ψ∗. It follows that ψ∗ ∈ Z 2(G , A). The 2-cocycle ψ∗ is called a 2-coboundary.
2-coboundaries form a subgroup B 2(G , A) of Z 2(G , A). We have proved:
Proposition 3.4.6 The extension A / µ / E ε / / G , where A is abelian, determines a
unique element φ + B 2(G , A) of the group Z 2(G , A)/B 2(G , A).
Choose a transversal function τ: G → E , i.e., τε = 1G . Then the above action can be
rewritten as
(a g )µ = g −τa µ g τ.
Let x , y ∈ G . As x τy τ and (x y )τ belong to the same coset of ker ε = im µ in E , we may
write
x τy τ = (x y )τ((x , y )φ)µ
for some (x , y )φ ∈ A. Thus we get a function φ : G × G → A defined by
((x , y )φ)µ = (x y )−τx τy τ.
From the associative law x τ(y τz τ) = (x τy τ)z τ we get that φ satisfies the identity
(x , y z )φ + (y , z )φ = (x y , z )φ + (x , y )φ · z .
A function φ : G × G → A satisfiying this functional equation is called a factor set (or a 2-
cocycle). Note that we can assume without loss of generality that 1τ = 1, therefore we can
always assume that (1, x )φ = (x , 1)φ = 0 for all x ∈ G . The set Z 2(G , A) of all 2-cocycles
in G with coefficients in the G -module A has the structure of an abelian group with the
operation
(x , y )(φ1 + φ2) = (x , y )φ1 + (x , y )φ2.
Example 3.4.5 In the situation above, what happens if (x , y )φ = 0 for all x , y ∈ G ? In this
case, the transversal map τ: G → E is a homomorphism. It is easy to see that the image of
τ is then a complement of im µ ∼= A in E , therefore E =∼ G χ A.
How does the choice of τ affect φ? Let τ be another transversal function for given
extension. Then we get another factor set φ , i.e., x τ y τ = (x y )τ ((x , y )φ )µ. As x τ and x τ
belong to the same coset of ker ε = im µ, we can write
x τ = x τ((x )ψ)µ
for some (x )ψ ∈ A. We get
(x , y )φ = (x , y )φ + (x y )ψ − (x )ψ · y − (y )ψ.
Define ψ∗ : G × G → A by
(x , y )ψ∗ = (y )ψ − (x y )ψ + (x )ψ · y ,
so that φ = φ+ψ∗. It follows that ψ∗ ∈ Z 2(G , A). The 2-cocycle ψ∗ is called a 2-coboundary.
2-coboundaries form a subgroup B 2(G , A) of Z 2(G , A). We have proved:
Proposition 3.4.6 The extension A / µ / E ε / / G , where A is abelian, determines a
unique element φ + B 2(G , A) of the group Z 2(G , A)/B 2(G , A).
