OAI.SourceBurnside.thm_main

The theorem states that two propositions hold together. For a natural number n and a ring R, a root is an ordered pair (i,j) of distinct indices in {0,…,n−1}, and the Steinberg group St(n,R) is the group generated by symbols x_{ij}(a), one for each root (i,j) and each a in R, subject to three families of relations: x_{ij}(a+b) = x_{ij}(a)x_{ij}(b); the commutator [x_{ij}(a), x_{kl}(b)] = x y x⁻¹ y⁻¹ is trivial whenever j≠k and i≠l; and for pairwise distinct i, j, k, the commutator [x_{ij}(a), x_{jk}(b)] equals x_{ik}(ab). A group is periodic if every element g has some positive integer m with gᵐ = 1. The first statement, MainStatement, asserts that there exist a ring R in the lowest universe, carrying an algebra structure over ZMod 2, such that St(12,R) is infinite, finitely presented, and periodic. The second statement, BurnsideStatement, asserts that there exists an infinite, finitely presented, periodic group G. The theorem asserts the conjunction of these two existence claims, with no bound on the orders of elements.