Volume 1 • Issue 2 • PP: 13-16 • 2022
On Some Results about Finite Non Simple Groups
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© 2022 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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This paper is dedicated to study some properties of finite groups, where we present the following results: 1) If all centralizers of a group G are cyclic, then G is not simple. 2) If the center of the subgroup A is not trivial and B is cyclic, then G=AB is not simple. On the other hand, we define the m-solvable groups, and we examine some of their elementary properties.
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References
[1]. Ashbacher . F-sets and permutation group . J. Algebra , 30N1-3 , 400,1974 .
[2]. Dayoub A. Study about supersoluble groups and some generalization for it Algebra M.Sc. Dissertation.2009 .
[3]. Glauberman G. Central elements in core – free groups . Ann.Math.99.N1,70,1974 .
[4]. Huppert B. Endliche Gruppen ,I. Berrlin – Heidel – berg –New York , 1967 .
[5]. Ito N. A theorem on factorizable groups . Acta Sci. math.,33,Nt2 , 49 , 1972 .
[6]. Ito N. On factorizable groups . Proc. Symp. Pire Math.,21,Represent . theoryfinite Relat . Top. Providence , R. 1,77,1971 .
[7]. Ito N. On the factorization of the linear factional group LF(2, ) , Acta scient . math. 15,79 ,1953
[8]. Janko Z. A new finite simple group with abelian Sylow 2-subgroups and its characterization .J. Algebra ,3,N2,147.1966.
[9]. Janko Z. , Thompson J. On a class of finite simple group of Ree.J. Algebra , 4 , N2, 274.1966.
[10]. Milne J. group theory Solomon Bull.Amer.,Math,Soc.,2007 .
[11]. Monakhov B. S. About prouduct two subgroubs , One is contain cyclic subgroup
with uidex 2. Math Zametki 16 N2 285 , 1974 (Russian)
[12]. Suzuki M. On a class of doubly transitive groups . I, Ann.Math., 75 , 105 , 1962 , 79 , 514 . 1964
[13]. Tchounikhin S.A. About development theory of finite group in leatest years .Uspech Math. Sci.16N4 1961 (Russian)
[14]. Tchounikhin S. A. About fineding sub group sin finite groups. Trudi semenar in group theory . M. L. 1938. (Russian)
[15] Sankari, H., and Abobala, M., On A New Criteria For the Solvability of Non-Simple Finite Groups and m-Abelian Solvability, Journal of Mathematics, Hindawi, 2021.
[16] Hatip, A., Alhamido, R., and Abobala, M., "A Contribution to Neutrosophic Groups", International Journal of Neutrosophic Science", Vol. 0, pp. 67-76 . 2019.
[17] Abobala, M., " n-Refined Neutrosophic Groups I", International Journal of Neutrosophic Science, Vol. 0, pp. 27-34. 2020.
[18] Abobala, M., " Analyzing On My Turf Game Over Some Finite Non-Abelian Groups", Galoitica Journal Of Mathematical Structures and Applications, 2022.
[19]. Tchounikhin S.A. Simplicite du groupe finite les ordres deses classes d'eléments conjugés Compt. Rend. Acad. Sci. Paris,
191 , 397. 1930 .
[20]. Tchounikhin S. A. subgroups of finite groups , Minsk 1964.(Russian)
[21]. Tchounikhin S.A. Sur le probleme des deux classes d'ungroupe fini. Compt. Rend. Acad. Sci. Paris, 198, 531, 1934 .
[22]. Walter J. The characterization of finite groups with abelian Sylow 2- sub groups . Ann. Math 89 N3 405. 1969 .
[23]. Ward H. On Ree's series of simple groups Trans. Ames.math. Soc. , 121 , N1 , 62 , 1966 .
[24]. Wielandt H. Finite permutation groups . New York –London ,1964 .
[25] Abobala, M. and Sankari, H. A Contribution to (m-Power Closed) Groups, UMM Al-Qura University Journal of Applied Science (UQUJAS), KSA, (2020).
[26]Von Shtawzen, O., " On A Novel Group Derived From A Generalization Of Integer Exponents And Open Problems", Galoitica Journal Of Mathematical Structures and Applications, 2022.
[27] Abobala, M., " A Study Of Novel Algebraic Game Over Some Finite Groups And Open Problems", Galoitica Journal Of Mathematical Structures and Applications, 2022.
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