 Transgression in group cohomology

For a group extension $1\to H\to G\to Q\to 1$ and a $G$-module $M$, the transgressionis the first spectral sequence differential crossing from the vertical edge to the horizontal edge of the Lyndon–Hochschild–Serre spectral sequence. By the five-term exact sequence in group cohomology, an invariant one-cocycle class on $H$ has zero transgression precisely when it is the restriction of a one-cocycle class on $G$. Thus transgression measures an extension obstruction. An arbitrary extension of a representative to a group cochain on $G$ need not have a coboundary that already descends to $Q$; the cochain construction requires appropriate corrections.

For $1\to C_2\to C_4\to C_2\to 1$ with trivial coefficients ${\mathbb{F}}_2$, every group homomorphism $C_4\to {\mathbb{F}}_2$ kills its order-two subgroup. Consequently the nonzero class in $H^1(C_2,{\mathbb{F}}_2)$ has nonzero transgression. It detects this nonsplit group extension, represented by the carry two-cocycle $c(a,b)=ab$ for $a,b\in \{0,1\}$.