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The Frattini p-subalgebra of a solvable Lie p-algebra

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The Frattini p-subalgebra of a solvable Lie p-algebra. / Lincoln, Mark; Towers, David.
In: Proceedings of the Edinburgh Mathematical Society, Vol. 40, No. 1, 1997, p. 31-40.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Lincoln, M & Towers, D 1997, 'The Frattini p-subalgebra of a solvable Lie p-algebra', Proceedings of the Edinburgh Mathematical Society, vol. 40, no. 1, pp. 31-40. https://doi.org/10.1017/S0013091500023415

APA

Lincoln, M., & Towers, D. (1997). The Frattini p-subalgebra of a solvable Lie p-algebra. Proceedings of the Edinburgh Mathematical Society, 40(1), 31-40. https://doi.org/10.1017/S0013091500023415

Vancouver

Lincoln M, Towers D. The Frattini p-subalgebra of a solvable Lie p-algebra. Proceedings of the Edinburgh Mathematical Society. 1997;40(1):31-40. doi: 10.1017/S0013091500023415

Author

Lincoln, Mark ; Towers, David. / The Frattini p-subalgebra of a solvable Lie p-algebra. In: Proceedings of the Edinburgh Mathematical Society. 1997 ; Vol. 40, No. 1. pp. 31-40.

Bibtex

@article{3c24eec36408491aaf814289d4d2ebdd,
title = "The Frattini p-subalgebra of a solvable Lie p-algebra",
abstract = "In this paper we continue our study of the Frattini p-subalgebra of a Lie p-algebra L. We show first that if L is solvable then its Frattini p-subalgebra is an ideal of L. We then consider Lie p-algebras L in which L^2 is nilpotent and find necessary and sufficient conditions for the Frattini p-subalgebra to be trivial. From this we deduce, in particular, that in such an algebra every ideal also has trivial Frattini p-subalgebra, and if the underlying field is algebraically closed then so does every subalgebra. Finally, we consider Lie p-algebras L in which the Frattini p-subalgebra of every subalgebra of L is contained in the Frattini p-subalgebra of L itself.",
author = "Mark Lincoln and David Towers",
note = "http://journals.cambridge.org/action/displayJournal?jid=PEM The final, definitive version of this article has been published in the Journal, Proceedings of the Edinburgh Mathematical Society, 40 (1), pp 31-40 1997, {\textcopyright} 1997 Cambridge University Press.",
year = "1997",
doi = "10.1017/S0013091500023415",
language = "English",
volume = "40",
pages = "31--40",
journal = "Proceedings of the Edinburgh Mathematical Society",
issn = "0013-0915",
publisher = "Cambridge University Press",
number = "1",

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RIS

TY - JOUR

T1 - The Frattini p-subalgebra of a solvable Lie p-algebra

AU - Lincoln, Mark

AU - Towers, David

N1 - http://journals.cambridge.org/action/displayJournal?jid=PEM The final, definitive version of this article has been published in the Journal, Proceedings of the Edinburgh Mathematical Society, 40 (1), pp 31-40 1997, © 1997 Cambridge University Press.

PY - 1997

Y1 - 1997

N2 - In this paper we continue our study of the Frattini p-subalgebra of a Lie p-algebra L. We show first that if L is solvable then its Frattini p-subalgebra is an ideal of L. We then consider Lie p-algebras L in which L^2 is nilpotent and find necessary and sufficient conditions for the Frattini p-subalgebra to be trivial. From this we deduce, in particular, that in such an algebra every ideal also has trivial Frattini p-subalgebra, and if the underlying field is algebraically closed then so does every subalgebra. Finally, we consider Lie p-algebras L in which the Frattini p-subalgebra of every subalgebra of L is contained in the Frattini p-subalgebra of L itself.

AB - In this paper we continue our study of the Frattini p-subalgebra of a Lie p-algebra L. We show first that if L is solvable then its Frattini p-subalgebra is an ideal of L. We then consider Lie p-algebras L in which L^2 is nilpotent and find necessary and sufficient conditions for the Frattini p-subalgebra to be trivial. From this we deduce, in particular, that in such an algebra every ideal also has trivial Frattini p-subalgebra, and if the underlying field is algebraically closed then so does every subalgebra. Finally, we consider Lie p-algebras L in which the Frattini p-subalgebra of every subalgebra of L is contained in the Frattini p-subalgebra of L itself.

U2 - 10.1017/S0013091500023415

DO - 10.1017/S0013091500023415

M3 - Journal article

VL - 40

SP - 31

EP - 40

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 1

ER -