Coprime splitting over an elementary abelian normal subgroup ($G=N⋊H,\gcd (|N|,|H|)=1$)

If $N$ is an elementary abelian normal subgroup and $|G/N|$ is prime to its characteristic, the extension splits and its complements are conjugate by $N$. In additive notation an extension cocycle satisfies $f(h,k)+f(hk,t)=hf(k,t)+f(h,kt)$. Averaging over $t$ gives $f(h,k)=b(h)+hb(k)-b(hk)$, where $b(h)=|G/N{|}^{-1}{\sum }_{t}f(h,t)$. Changing the section by $b$ removes the cocycle. For two complements the difference cocycle has $d(hk)=d(h)+hd(k)$; averaging gives $d(h)=b-hb$, which is conjugation by an element of $N$. These arguments use invertibility of the group order on $N$ and underlie Hall conjugacy in soluble groups.