Finitely presented infinite p-groups for every prime

For every prime p we construct an infinite, residually finite group with an ordinary finite presentation in which every element has finite p-power order. For p = 2 this is a recent construction of OpenAI, which also answered the finitely presented Burnside question. We extend it to every prime, using a single finite presentation written over the integers. Over every field of positive characteristic this presentation defines a finitely presented, infinite-dimensional, nil, Jacobson radical algebra that is not nilpotent and whose matrix algebras are all nil. Reducing it modulo p gives a graded F_p-algebra R_p, and the groups are finite-index subgroups of the Steinberg groups St_12(R_p). The extension is not a relabelling of 2 by p. The characteristic-two cancellation bound must be replaced by one that depends on p. The unordered-frame homological stability argument fails over odd fields and is replaced by an ordered-frame argument with signed transpositions. The phases that detect central elements must be taken modulo integers coprime to p. Finally, we show that characteristic zero is genuinely different: over a field of characteristic zero, an algebra presented by finitely many word equalities and word zeros is nil only if it is nilpotent. A characteristic-zero analogue therefore needs relations with signs or coefficients.