= Maximal subgroups of a finite soluble group
{title2=$G=AB\quad\text{or}\quad A^g=B$}
Two maximal subgroups of a finite soluble group either multiply to the whole group or are conjugate. Reduce modulo a nontrivial normal subgroup contained in either maximum. If no such subgroup exists, both maxima complement a minimal normal elementary abelian subgroup $K$. Their actions on $K$ are faithful and irreducible. A minimal normal subgroup of $G/K$ has characteristic different from that of $K$, since a normal p-subgroup has nonzero fixed points on a characteristic-p module. The intersections of the two maxima with its preimage are conjugate Sylow subgroups. Each maximum is the normalizer of its nontrivial intersection, proving conjugacy.
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