= Finite p-groups are nilpotent
Induct on the order of the finite <p-group>. Its <center of a group> is nontrivial unless the group itself is trivial. Quotient by the center to obtain a smaller finite p-group and lift a <central series> supplied by induction. Prepending the center gives a central series for the original group: the first term is central, and the remaining commutator-containment conditions are exactly the lifted quotient conditions. Hence every finite p-group is a <nilpotent group>. Since central factors are Abelian, every nilpotent group is also a <soluble group>.
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