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Let G be a finite p-separable group, for some fixed prime p. Let Γ p (G) be the common divisor graph built on the set of non-central conjugacy classes of p-regular elements of G: this is the graph whose vertices are the conjugacy classes of those non-central elements of G such that p does not divide their orders, and two distinct vertices are adjacent if and only if the greatest common divisor of their lengths is strictly greater than one. The aim of this paper is twofold: to positively answer an open question concerning the maximum possible distance in Γ p (G) between a vertex with maximal cardinality and any other vertex, and to study the p-structure of G when Γ p (G) has diameter three. [ABSTRACT FROM AUTHOR]

G-character tables of a finite group G were defined in Felipe et al. (Quaest Math, 2022. ). These tables can be very useful to obtain certain structural information of a normal subgroup from the character table of G. We analyze certain structural properties of normal subgroups which can be determined using their G-character tables. For instance, we prove an extension of the Thompson’s theorem from minimal G-invariant characters of a normal subgroup. We also obtain a variation of Taketa’s theorem for hypercentral normal subgroups considering their minimal G-invariant characters. This generalization allows us to introduce a new class of nilpotent groups, the class of nMI-groups, whose members verify that its nilpotency class is bounded by the number of irreducible character degrees of the group. [ABSTRACT FROM AUTHOR]