Generalization of some classical results of Srinivasan

• Published in: Archiv der Mathematik Vol. 119; no. 1; pp. 11 - 18
• Main Authors: Qiao, Shouhong, Cao, Chenchen, Liu, A-Ming, Guo, W.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2022; Springer Nature B.V
• Subjects: Group theory; Mathematics; Mathematics and Statistics; Prime numbers; Subgroups; 20D10; 20D20; Supersoluble groups; normal subgroups; permutable subgroups; subnormal subgroups; soluble groups
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-022-01750-0

Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers, G a finite group, π ( G ) the set of all primes dividing | G |, and σ ( G ) = { σ i : σ i ∩ π ( G ) ≠ ∅ } . The group G is called a σ -group if G has a set of subgroups H such that every non-trivial subgroup contained in H is a Hall σ i -subgroup of G and H contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ ( G ) . In this paper, we investigate the structure of the σ -groups G by using the σ -normality, σ -permutability, and σ -subnormality of maximal subgroups of elements in H . Some criteria of supersolubility and σ -solubility of G are obtained, which generalize some classical results of Srinivasan.