in the presence of a large group action, even non-noetherian rings sometimes behave like noetherian rings. for instance, cohen proved that every symmetric ideal in the infinite variable polynomial ring is generated by the symmetric group orbit of finitely many polynomials. in this talk, i will provide a brief introduction to gl-algebras—commutative algebras equipped with an action of the infinite general linear group—and recall the rich history of the noetherianity problem for gl-algebras. i will then present recent work where i construct a counterexample to the algebraic noetherianity problem for gl-algebras in characteristic two.