Geodesic
SCAP-subalgebras of Lie algebras
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1177-1184
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A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $\rm SCAP$-subalgebra if there is a chief series $0=L_0\subset L_1\subset \ldots \subset L_t=L$ of $L$ such that for every $i=1,2,\ldots ,t$, we have $H+L_i=H+L_{i-1}$ or $H\cap L_i=H\cap L_{i-1}$. This is analogous to the concept of $\rm SCAP$-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $\rm SCAP$-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.
A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $\rm SCAP$-subalgebra if there is a chief series $0=L_0\subset L_1\subset \ldots \subset L_t=L$ of $L$ such that for every $i=1,2,\ldots ,t$, we have $H+L_i=H+L_{i-1}$ or $H\cap L_i=H\cap L_{i-1}$. This is analogous to the concept of $\rm SCAP$-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $\rm SCAP$-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.
DOI : 10.1007/s10587-016-0317-3
Classification : 17B05, 17B30, 17B50
Keywords: Lie algebra; $\rm SCAP$-subalgebra; chief series; solvable; supersolvable 
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Chehrazi, Sara; Salemkar, Ali Reza. SCAP-subalgebras of Lie algebras. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 4, pp. 1177-1184. doi: 10.1007/s10587-016-0317-3

[1] Ballester-Bolinches, A., Ezquerro, L. M., Skiba, A. N.: On second maximal subgroups of Sylow subgroups of finite groups. J. Pure Appl. Algebra 215 (2011), 705-714. | DOI | MR | Zbl

[2] Barnes, D. W.: On Cartan subalgebras of Lie algebras. Math. Z. 101 (1967), 350-355. | DOI | MR | Zbl

[3] Barnes, D. W.: On the cohomology of soluble Lie algebras. Math. Z. 101 (1967), 343-349. | DOI | MR | Zbl

[4] Borel, A., Mostow, G. D.: On semi-simple automorphisms of Lie algebras. Ann. Math. (2) 61 (1955), 389-405. | DOI | MR | Zbl

[5] Fan, Y., Guo, X. Y., Shum, K. P.: Remarks on two generalizations of normality of subgroups. Chin. Ann. Math. Ser. A 27 (2006), 169-176. | MR | Zbl

[6] Graaf, W. A.: Lie Algebras: Theory and Algorithms. North-Holland Mathematical Library 56 North-Holland, Amsterdam (2000). | MR | Zbl

[7] Guo, X., Wang, J., Shum, K. P.: On semi-cover-avoiding maximal subgroups and solvability of finite groups. Comm. Algebra 34 (2006), 3235-3244. | DOI | MR | Zbl

[8] Hallahan, C. B., Overbeck, J.: Cartan subalgebras of meta-nilpotent Lie algebras. Math. Z. 116 (1970), 215-217. | DOI | MR | Zbl

[9] Li, Y., Miao, L., Wang, Y.: On semi cover-avoiding maximal subgroups of Sylow subgroups of finite groups. Commun. Algebra 37 (2009), 1160-1169. | DOI | MR | Zbl

[10] Salemkar, A. R., Chehrazi, S., Tayanloo, F.: Characterizations for supersolvable Lie algebras. Commun. Algebra 41 (2013), 2310-2316. | DOI | MR | Zbl

[11] Stitzinger, E. L.: On the Frattini subalgebra of a Lie algebra. J. Lond. Math. Soc. 2 (1970), 429-438. | DOI | MR | Zbl

[12] Stitzinger, E. L.: Covering-avoidance for saturated formations of solvable Lie algebras. Math. Z. 124 (1972), 237-249. | DOI | MR | Zbl

[13] Towers, D.: Lie algebras all of whose maximal subalgebras have codimension one. Proc. Edinb. Math. Soc. (2) 24 (1981), 217-219. | DOI | MR | Zbl

[14] Towers, D. A.: c-ideals of Lie algebras. Commun. Algebra 37 (2009), 4366-4373. | DOI | MR | Zbl

[15] Towers, D. A.: Subalgebras that cover or avoid chief factors of Lie algebras. Proc. Am. Math. Soc. 143 (2015), 3377-3385. | DOI | MR

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