University of KashanMathematics Interdisciplinary Research2538-36395420201201n-capability of A-groups34535311082110.22052/mir.2020.240334.1251ENMarziehChakanehDepartment of Pure Mathematics,
Ferdowsi University of Mashhad,
P. O. Box 1159, Mashhad 91775, I. R. IranSaeedKayvanfarDepartment of Pure Mathematics,
Ferdowsi University of Mashhad,
P. O. Box 1159, Mashhad 91775, I. R. IranRasoulHatamianDepartment of Pure Mathematics,
Payame Noor University,
Tehran, I. R. IranJournal Article20201022Following P. Hall a soluble group whose Sylow subgroups are all abelian is called A-group. The purpose of this article is to give a new and shorter proof for a criterion on the capability of A-groups of order p<sup>2</sup>q, where p and q are distinct primes. Subsequently we give a sufficient condition for n-capability of groups having the property that their center and derived subgroups have trivial intersection, like the groups with trivial Frattini subgroup and A-groups. An interesting necessary and sufficient condition for capability of the A-groups of square free order will be also given.https://mir.kashanu.ac.ir/article_110821_0abe1545736a161f9fd287d64f272d75.pdf