= Hall conjugacy and embedding in finite soluble groups
{c}
= Hall theorem for soluble groups
{c}
{synonym}
For any prime set $\pi$, every finite soluble group has Hall $\pi$-subgroups, any two are conjugate, and every $\pi$-subgroup lies in one. Induction uses an elementary abelian minimal <normal subgroup> $N$ of characteristic $p$. If $p\in\pi$, the preimage of a quotient Hall <subgroup> is itself a Hall <subgroup>. If $p\notin\pi$, <coprime splitting over an elementary abelian normal subgroup> gives a conjugate class of complements in that preimage. The same complement conjugacy in the preimage of a smaller quotient <subgroup> gives the embedding assertion.
Back to article page
