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

Published online by Cambridge University Press:  20 January 2009

Mark Lincoln
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, England
David Towers
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, England
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Abstract

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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 L2 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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1997

References

REFERENCES

1. Barnes, D. W., Lie algebras, (Lecture notes, University of Tubingen, 1969).Google Scholar
2. Jacobson, N., Lie algebras, (Interscience, New York, 1962).Google Scholar
3. Lincoln, M. and Towers, D. A., Frattini theory for restricted Lie algebras, Arch. Math. 45 (1985), 451457.CrossRefGoogle Scholar
4. Schue, J. R., Cartan decompositions for Lie algebras of prime characteristic, J. Algebra 11 (1969), 2552.CrossRefGoogle Scholar
5. Stitzinger, E. L., Frattini subalgebras of a class of solvable Lie algebras, Pacific J. Math. 34 (1970), 177182.CrossRefGoogle Scholar
6. Towers, D. A., A Frattini theory for algebras, Proc. London Math. Soc. (3) 27 (1963), 440462.Google Scholar
7. Winter, D. J., Abstract Lie algebras, (MIT Press, 1972).Google Scholar