On Boundary Factors and Traces of Subgroups of Finite Groups

Abstract

A subgroup $E$ of a finite group $G$ is called hypercyclically embedded in $G$ if every chief factor of $G$ below $E$ is cyclic. Let $A$ be a subgroup of a group $G$. Then we call any chief factor $H/A_{G}$ of $G$ a $G$-boundary factor of $A$. For any $G$-boundary factor $H/A_{G}$ of $A$, we call the subgroup $(A\cap H)/A_{G}$ of $G/A_{G}$ a $G$-trace of $A$. On the basis of these notions, we give some new characterizations of hypercyclically embedded subgroups.

Keywords

Finite group / Hypercyclically embedded subgroup / $G$-boundary factor')">$G$-boundary factor / $G$-trace of subgroup')">$G$-trace of subgroup / Meet-irreducible subgroup

References

[1] Johnson DL. A note on supersoluble groups. Canadian J. Math.. 1971, 23 562-564

[2] Weinstein M. Between Nilpotent and Solvable. 1982 Passaic: Polygonal Publishing House

[3] Birkhoff G. Lattices Theory. 1984 3 Providence: AMS Coll. Publ.

[4] Schmidt R. Subgroup Lattices of Groups. 1994 Berlin: Walter de Gruyter

[5] Ballester-Bolinches A, Esteban-Romero R, Asaad M. Products of Finite Groups. 2010 Berlin-New York: Walter de Gruyter

[6] Shemetkov LA, Skiba AN. On the ${{\cal X}\Phi } $-hypercentre of finite groups. J. Algebra. 2009, 322 2106-2117

[7] Asaad M. Finite groups with certain subgroups of Sylow subgroups complemented. J. Algebra. 2010, 323 7 1958-1965

[8] Skiba AN. On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups. J. Group Theory. 2010, 13 841-850

[9] Li B. On $\Pi $-property and $\Pi $-normality of subgroups of finite groups. J. Algebra. 2011, 334 321-337

[10] Skiba AN. A characterrization of hypercyclically embedded subgroups of finite groups. J. Pure Appl. Algebra. 2011, 215 257-261

[11] Guo W, Skiba AN. On ${{\cal F}\Phi ^{*}}$-hypercentral subgroups of finite groups. J. Algebra. 2012, 372 275-292

[12] Yi X, Alexander N. Skiba, some new characterizations of PST-groups. J. Algebra. 2014, 399 39-54

[13] Chen X, Guo W, Skiba AN. Some conditions under which a finite group belongs a Baer local formation. Comm. Algebra. 2014, 42 4188-4205

[14] Doerk K, Hawkes T. Finite Soluble Groups. 1992 Berlin-New York: Walter de Gruyter

[15] Skiba AN. Finite groups with given systems of generalized permutable subgroups. Proc. F.Scorina Gomel State Univ.. 2006, 36 3 12-31

[16] Ballester-Bolinches A, Ezquerro LM, Skiba AN. Local embeddings of some families of subgroups of finite groups. Acta Math. Sin.. 2009, 25 869-882

[17] Guo X, Wang J, Shum KP. On semi-cover-avoiding maximal subgroups and solvability of finite groups. Comm. Algebra. 2006, 34 3235-3244

[18] Huppert B. Endliche Gruppen I. 1967 Berlin, Heidelberg, New York: Springer-Verlag

[19] Deskins WE. On maximal subgroups. Proc. Symp. Pure Math., Am. Math. Soc. 1959, 1 100-104

[20] Knyagina BN, Monakhov VS. On ${\pi ^{\prime }}$-properties of finite groups having a Hall $\pi $-subgroup. Sib. Math. J.. 2011, 522 309-398

[21] Huppert B, Blackburn N. Finite Groups III. 1982 Berlin-New York: Springer-Verlag

[22] Gorenstein D. Finite Groups. 1968 New York, Evanston, London: Harper & Row Publishers

[23] Guo X, Shum KP. Cover-avoidance properties and the structure of finite groups. J. Pure Appl. Algebra. 2003, 181 297-308

[24] Kurzweil H, Stellmacher B. The Theory of Finite Groups: An Introduction. 2004 New York-Berlin-Heidelberg: Springer-Verlag