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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 of socle-regularity. We show that large classes of p-groups, including all divisible, totally projective and torsion-complete p-groups, share this property when the prime p is odd. The present work generalizes notions of full inertia intensively studied recently by several authors and is a development of a recent work of the authors published in Mediterranean J. Math. (2021).
Chekhlov, A., Danchev, P. & Goldsmith, B. (2021). On the socles of characteristically inert subgroups of Abelian p-groups. Forum Mathematicum, 33(4), 889-898. DOI: 10.1515/forum-2020-0348