Some Results on the Partition of Groups

Abstract

We investigate some properties of partitions of groups. Some information on the structure of nilpotent groups as well the embedding of groups with nontrivial partitions into a larger one are discussed. Moreover, we consider the structure of groups admitting a nontrivial partition with 1, 2 or 3 subgroups (considering the whole group) with nontrivial partition and the relation between the number of subgroups with nontrivial partition and the number of primes dividing the order of the group. Finally we analyze the relation between the number of components of a nontrivial partition of a group and its order.