Deciphering subgroups of direct products

Stephanos Gekas

Published 2015-09-10Version 1

In this article we are dealing with the general problem of investigating the subgroups of a direct product of groups. What is most innovative in this approach is our method, i.e. a suitably defined equivalence relation, which is pecisely what allows for treating the quite hard and confusing case of multiple factors, thus giving a new and deeper insight into the structure of such subgroups. We explore the implementation of that relation first in the (already well known) case of two factors, to make the (non-trivial!) generalizations to n factors afterwards and obtain clear results for the general case too, summarized in theorems 5, 20 and 29. Some applications and new techniques arizing from this approach are examined, most notably concerning basic properties of homomorphisms, extending well-known elementary ones.