On the global breadth of finite groups with nontrivial partitions

In a series of recent contributions on the notion of global breadth B (G) of a finite group G, it was interesting to observe the structural conditions arising from the classification of finite groups of B (G) =8. This motivated the study of a new class of finite groups, namely H=\G \ | \ G \ satisfies the condition \ |G| B (G) (B (G) + 1) \ and very little is known about H. Here we focus on the groups with nontrivial partitions (according to the terminology of Baer, Kegel and Kontorovich), determining first that B (G) is achieved via the local breadth in connection with the order of maximal cyclic subgroups. Then we show that H contains projective special linear groups, projective general linear groups and Suzuki groups, supporting the conjecture that all finite groups with nontrivial partitions belong to H. The presence of large families of simple groups in H is shown for the first time here.