On weakly π-permutable subgroups of finite groups
-
Chenchen Cao
,
Venus Amjid
Abstract
Let Ο = {Οi β£i β I} be some partition of the set of all primes β, G be a finite group and Ο(G) = {Οiβ£Οi β© Ο(G) β β }. G is said to be Ο-primary if β£Ο(G)β£ β€ 1. A subgroup H of G is said to be Ο-subnormal in G if there exists a subgroup chain H = H0 β€ H1 β€ β¦ β€ Ht = G such that either Hiβ1 is normal in Hi or Hi/(Hiβ1)Hi is Ο-primary for all i = 1, β¦, t. A set π of subgroups of G is said to be a complete HallΟ-set of G if every non-identity member of π is a Hall Οi-subgroup of G for some i and π contains exactly one Hall Οi-subgroup of G for every Οi β Ο(G). Let π be a complete Hall Ο-set of G. A subgroup H of G is said to be π-permutable if HA = AH for all A β π. We say that a subgroup H of G is weakly π-permutable in G if there exists a Ο-subnormal subgroup T of G such that G = HT and H β© T β€ Hπ, where Hπ is the subgroup of H generated by all those subgroups of H which are π-permutable.
By using the weakly π-permutable subgroups, we establish some new criteria for a group G to be Ο-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.
This work was supported by the NNSF of China (11771409), Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences and Anhui Initiative in Quantum Information Technologies (AHY150200)
(Communicated by Vincenzo Marra )
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Articles in the same Issue
- Regular papers
- On the finite embeddability property for quantum B-algebras
- A dual Ramsey theorem for finite ordered oriented graphs
- On EMV-Semirings
- A nonsymmetrical matrix and its factorizations
- On weakly π-permutable subgroups of finite groups
- The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions
- A geometrical version of Hardy-Rellich type inequalities
- On a Choquet-Stieltjes type integral on intervals
- Uniqueness of meromorphic functions sharing four small functions on annuli
- A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator
- Functions of bounded variation related to domains bounded by conic sections
- Unified solution of Fekete-SzegΓΆ problem for subclasses of starlike mappings in several complex variables
- Oscillatory criteria for the second order linear ordinary differential equations
- When deviation happens between rough statistical convergence and rough weighted statistical convergence
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- On sequence spaces generated by binomial difference operator of fractional order
- Some refinements of young type inequality for positive linear map
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