= Coprime splitting over an elementary abelian normal subgroup
{title2=$G=N\rtimes H,\quad\gcd(|N|,|H|)=1$}
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 $f(h,k)+f(hk,t)=h f(k,t)+f(h,kt)$. Averaging over $t$ gives $f(h,k)=b(h)+h b(k)-b(hk)$, 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 $d(hk)=d(h)+h d(k)$; 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.
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