On Fitting groups whose proper subgroups are solvable

Document Type : Research Paper

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Abstract

This work is a continuation of [A‎. ‎O‎. ‎Asar‎, ‎On infinitely generated groups whose proper subgroups are solvable‎,  J‎. ‎Algebra, ‎399 (2014) 870-886.]‎, ‎where it was shown‎ ‎that a perfect infinitely generated group whose proper subgroups‎ ‎are solvable and in whose homomorphic images normal closures of‎ ‎finitely generated subgroups are residually nilpotent is a Fitting‎ ‎$p$-group for a prime $p$‎. ‎Thus this work is a study of a Fitting‎ ‎$p$-group whose proper subgroups are solvable‎. ‎New‎ ‎characterizations and some sufficient conditions for the‎ ‎solvability of such a group are obtained‎.

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A‎. ‎Arikan and H‎. ‎Smith (2011). ‎On groups with all proper‎ ‎subgroups of finite exponent. J‎. ‎Group Theory. 14, 765-775 A‎. ‎O‎. ‎Asar (2000). ‎Locally nilpotent $p$-groups whose proper‎ ‎subgroups are hypercentral or nilpotent-by-Chernikov. J‎. ‎London Math‎. ‎Soc‎. ‎(2). 61, 412-422 A‎. ‎O‎. ‎Asar (2011). ‎Totally imprimitive permutation groups with the cyclic-block property. J‎. ‎Group Theory. 14, 127-141 A‎. ‎O‎. ‎Asar (2014). ‎On infinitely generated groups whose proper subgroups are solvable. J‎. ‎Algebra. 399, 870-886 G‎. ‎Endimioni and G‎. ‎Traustason (2001). ‎On torsion-by-nilpotent groups. J‎. ‎Algebra. 241, 669-676 D‎. ‎Gorenstein (1968). Finite Groups. ‎Harper \& Row‎, ‎Publishers‎, ‎New York-London. B‎. ‎Huppert (1967). Endliche Gruppen I. ‎Springer-Verlag‎, ‎Berlin‎, ‎Heidelberg‎, ‎New York. B‎. ‎Huppert and N‎. ‎Blackburn (1982). Finite Groups II. ‎Springer-Verlag‎, ‎Berlin‎, ‎Heidelberg‎, ‎New York. O‎. ‎H‎. ‎Kegel and B‎. ‎A‎. ‎F‎. ‎Wehrfritz (1973). Locally Finite Groups. ‎North-Holland Publishing Co.‎, ‎Amsterdam-London‎, ‎American Elsevier Publishing Co.‎, ‎Inc.‎, ‎New York. J‎. ‎C‎. ‎Lennox and D‎. ‎J‎. ‎S‎. ‎Robinson (2004). The Theory of Infinite Soluble Groups. ‎The Clarendon Press‎, ‎Oxford University Press‎, ‎Oxford. V‎. ‎D‎. ‎Mazurov and E‎. ‎I‎. ‎Khukhro (2014). Unsolved Problems in Group Theory. ‎(The Kourovka Notebook)‎, ‎Russian Academy of Sciences‎, ‎Siberian Division‎, ‎Novosibirsk. D‎. ‎J‎. ‎S‎. ‎Robinson (1972). Finiteness Conditions and Generalized Solvable Groups Part I‎, ‎II. ‎Springer‎, ‎New York. D‎. ‎J‎. ‎S‎. ‎Robinson (1980). A Course in the Theory of Groups. ‎Springer-Verlag‎, ‎New York‎, ‎Heidelberg‎, ‎Berlin. P‎. ‎Shumyatsky (2003). ‎On extensions of groups of finite exponent. Glasg‎. ‎Math‎. ‎J.. 45, 535-538