Cancellation of groups with maximal condition

Abstract:

It is not true that a group which obeys the maximal condition for normal subgroups may always be cancelled in direct products. However, we show the following Theorem. Let $C$ be a group which obeys the maximal condition for normal subgroup. Suppose further that if ${C_{\ast }}$ is an arbitrary homomorphic image of $C$, then ${C_{\ast }}$ is not isomorphic to a proper normal subgroup of itself. Then $C$ may be cancelled in direct products. Some generalizations of this result are indicated.