Maximal ideals in modular group algebras of the finitary symmetric and alternating groups

The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (pr ovided the characteristic p of the ground field is greater than 2). For the symmetric group there are exactly p-1 such ideals and for the alternating group there are (p-1)/2 of them. The description is obtained in terms of th e annihilators of certain systems of the 'completely splittable' irreducibl e modular representations of the finite symmetric and alternating groups. T he main tools used in the proofs are the modular branching rules (obtained earlier by the second author) and the 'Mullineux conjecture' proved recentl y by Ford-Kleshchev and Bessenrodt-Olsson. The results obtained are relevan t to the theory of PI-algebras. They are used in a later paper by the autho rs and A. E. Zalesskii on almost simple group algebras and asymptotic prope rties of modular representations of symmetric groups.