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- W2050874539 abstract "Theorems of Gaschiutz and Baer concerning the Frattini subgroup, the hypercenter, and nilpotent and supersolvable subgroups are extended to normally closed saturated formations. The hypercenter, derived group, and Frattini subgroup are all related to nilpotency of a finite group. In [4], Gaschiitz studied their relations, and found a sufficient condition for a normal subgroup of a group to be nilpotent. By replacing the notion of centrality with a-centrality, we give theorems which extend the results of Gaschuitz to normally closed saturated formations containing all nilpotent groups, and strengthen his nilpotency criterion. A theorem of Baer, on supersolvably immersed subgroups, is also extended to saturated formations. In this paper, all groups are finite. A class of groups is understood to include, with G, all groups isomorphic to G. If M is a subgroup of G, core(M) is the intersection of the G-conjugates to M, and is the largest normal subgroup of G contained in M. A chieffactor of G is a quotient H/K, where K< G and H/K is minimal normal in GIK. If H/K is a chief factor with L<K and H<M, we say H/K lies above L and below M. A subgroup M< G is a supplement to H/K in G if MH= G and MrH? K. If MnH=K, M is a complement to H/K. Other notation and terminology is standard, and can be found in [6]. Although the results below concern classes of solvable groups, it is not assumed that all groups dealt with are solvable. 1. Let X be a class of groups. A maximal subgroup M of G is X-normal if G/core(M) is in X, and X-abnormal otherwise. We define TX(G) to be the intersection of G with all X-normal maximal subgroups of G. If X is the class of all groups, TI(G) is the Frattini subgroup, 4D(G). Many useful properties of 4D(G) apply to TI(G); we collect them here. Received by the editors March 29, 1971. AMS 1970 subject classifications. Primary 20DI0, 20D25, 20D35." @default.
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- W2050874539 date "1972-02-01" @default.
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- W2050874539 title "Normally closed saturated formations" @default.
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- W2050874539 doi "https://doi.org/10.1090/s0002-9939-1972-0291271-6" @default.
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