On \(D(j)\)-groups with an element of order \(p^{j+1}\) for some prime \(p\) (Q6608142)

Let \(G\) be a group. A non-identity element \(x\) of \(G\) is called deficient if \(C_G(x)\neq\langle x\rangle\) and non-deficient if \(C_G(x)=\langle x\rangle\). Let \(j\) be a non-negative integer. The group \(G\) has defect \(j\), if \(G\) has exactly \(j\) deficient conjugacy classes. The authors of this paper introduced these notions in their previous work [J. Algebra 637, 112--131 (2024; Zbl 1527.20043)], where, among other results, they determined all finite groups with defect \(0\) and \(1\) and proved that a group which is either locally graded with defect \(0\) or locally graded periodic with defect \(1\) is finite. A group is locally graded if every nontrivial finitely generated subgroup has a proper subgroup of finite index.\N\NIn the paper under review, the authors determined all finite groups with defect \(j\geq1\) that contain an element of order \(p^{j+1}\) for some prime \(p\).\N\NIn addition, the authors proved the following two interesting theorems.\N\NTheorem (Theorem 3.1 and Theorem 3.2 in the paper). If \(j\geq1\) and \(G\) is a locally finite (or locally graded) group with defect \(j\) and \(G\) contains an element of order \(p^{j+1}\) for some prime \(p\), then \(G\) is a finite group.\N\NFinally, we note the following dedication by the authors: ``Dedicated to the memory of our colleague and friend Professor Nikolai Vavilov''.