Finite groups in which σ-quasinormality is a transitive relation

Let sigma = { sigma( i) | i is an element of I} be some partition of the set of all primes. A subgroup A of a finite group G is said to be: (i) sigma-subnormal in G if there is a subgroup chain A = A (0) <= A (1) <= center dot center dot center dot <= A (n) = G such that either A( i - 1) (sic) A (i) or A( i) / ( A( i - 1 )) A (i) is a sigma( j)-group, j = j( i), for all i = 1 , ... , n; (ii) modular in G if the following conditions are held: (1) < X, A boolean AND Z) = < X, A >boolean AND Z for all X <= G, Z <= G such that X <= Z, and (2) < A, Y boolean AND Z > = < A, Y >boolean AND Z for all Y <= G, Z <= G such that A <= Z; (iii) sigma-quasinormal in G if A is sigma-subnormal and modular in G. We obtain a description of finite groups in which sigma-quasinormality (respectively, modularity) is a transitive relation. Some known results are extended. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.