UvA

(Abstract of results (paraphrased): The paper classifies the finite
groups mentioned in the title. It turns out that the class of all finite
solvable groups satifying the condition in the title, coincides with the
class of all finite solvable groups satisfying the condition in the title
without the word "abelian" being printed. The last mentioned class of
groups was classified in the beginning of the 90-ties by van der Waall and
Ben Said, as well as the class of all the non-solvable finite groups
without the word "abelian" in the title. In the paper under consideration ,
the class of all non-solvable finite groups happens to be larger than the
class of the Van der Waall-Ben Said counterparts; Janko's first simple
group and some projective linear groups and linear groups over some finite
fields are involved now.)