an:05912826
Zbl 1230.20015
Guo, W.; Yu, X.
On \(\mathfrak F_n\)-normal subgroups of finite groups
EN
Sib. Math. J. 52, No. 2, 197-206 (2011); translation from Sib. Mat. Zh. 52, No. 2, 250-264 (2011).
00280593
2011
j
20D10 20D15 20D40
finite groups; supersoluble groups; saturated formations; \(p\)-nilpotent groups; supersolubility; \(p\)-nilpotency; supplemented subgroups; c-normal subgroups; hypercenter
All groups considered are finite. For a class \(\mathfrak F\) of groups, in the paper under review a subgroup \(H\) of a group \(G\) is defined to be \(\mathfrak F_n\)-normal in \(G\) if there exists a normal subgroup \(T\) of \(G\) such that \(HT\) is a normal subgroup of \(G\) and \((H\cap T)H_G/H_G\) is contained in the \(\mathfrak F\)-hypercenter \(Z_\infty^{\mathfrak F}(G/H_G)\) of \(G/H_G\), where \(H_G\) denotes the core of \(H\) in \(G\). We recall that the \(\mathfrak F\)-hypercenter is the largest normal subgroup of \(G\) whose \(G\)-chief factors are \(\mathfrak F\)-central (a chief factor \(H/K\) of \(G\) is \(\mathfrak F\)-central if \([H/K](G/C_G(H/K))\in\mathfrak F\)).
This concept is a generalization of \(c\)-normality (introduced by \textit{Y. Wang} [in J. Pure Appl. Algebra 110, No. 3, 315-320 (1996; Zbl 0853.20015)]), \(\mathfrak F_n\)-supplementation (or \(\mathfrak F_c\)-normality) (considered by \textit{N. Yang} and the first author [in Asian-Eur. J. Math. 1, No. 4, 619-629 (2008; Zbl 1176.20018)] and by \textit{A. Y. Alsheik Ahmad, J. J. Jaraden} and \textit{A. N. Skiba} [in Algebra Colloq. 14, No. 1, 25-36 ( 2007; Zbl 1126.20012)]) and \(\mathfrak F_h\)-normality (defined by \textit{X. Feng} and the first author [in Front. Math. China 5, No. 4, 653-664 (2010; Zbl 1226.20011)]).
In this paper the authors use this new embedding property to obtain some criteria for supersolubility and \(p\)-nilpotency (\(p\) a prime) of groups, taking further some previous developments.
Ana Mart??nez-Pastor (Valencia)
Zbl 0853.20015; Zbl 1176.20018; Zbl 1126.20012; Zbl 1226.20011