On the socles of characteristically inert subgroups of Abelian <i>p</i>-groups

Abstract We define the notion of a characteristically inert socle-regular Abelian p -group and explore such groups by focussing on their socles, thereby relating them to previously studied notions socle-regularity. show that large classes -groups, including all divisible, totally projective torsion-complete share this property when prime is odd. The present work generalizes full inertia intensively recently several authors development recent published in Mediterranean J. Math. (2021).