ABSTRACT
This chapter studies the extending structures (ES) problem for groups, i.e. given a group H and a set E such that H is a subset of E, the set of all group structures that can be defined on E such that H becomes a subgroup of E are described and classified up to an isomorphism that stabilizes H. It is shown that any such group structure on the set E, called unified product, is completely determined by a system of maps subject to certain compatibility conditions. The ES problem generalizes and unifies two famous problems in group theory: the extension problem of Hölder [133] and the factorization problem of Ore [192]. The main application refers to the classifying complements problem: for a sugbroup A of a given group G, describle and classify all complements of A in G (i.e. all subgroups H of G such that G=A H and A and H have trivial intersection). After providing the full solution to the above problem we obtain, as an application, a combinatorial formula for computing the number of isomorphisms types of groups of order n wich arises only from the exact factorization Sn=Sn− Cn.
