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The Frattini argument for Lie algebras. (English) Zbl 0253.17003

The Frattini argument shows that, if \(N\) is a normal subgroup of a group \(G\) and if \(S\) is an intravariant subgroup of \(N\), then \(G=\mathcal N_G(S)N\). Unfortunately, there appears to be no completely satisfactory Lie algebra analogue of the concept of an intravariant subgroup. We get around this difficulty by considering ways in which intravariant subgroups arise, and investigating the analogous situations for Lie algebras.
We show, subject to suitable conditions on \(N\) and \(\mathfrak X\) that if \(N\) is an ideal of the Lie algebra \(L\) and \(S\) is an \(\mathfrak X\)-projector of \(N\), then \(L=N+\mathcal N_L(S)\). In particular, this holds if \(S\) is a Cartan subalgebra of \(N\).
Reviewer: Donald W. Barnes

MSC:

17B05 Structure theory for Lie algebras and superalgebras
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References:

[1] Barnes, D.W.: On the cohomology of soluble Lie algebras. Math. Z.101, 343-349 (1967) · Zbl 0166.04102 · doi:10.1007/BF01109799
[2] Barnes, D.W., Gastineau-Hills, H.M.: On the theory of soluble Lie algebras. Math. Z.106, 343-354 (1968) · Zbl 0164.03701 · doi:10.1007/BF01115083
[3] Barnes, D.W., Newell, M.L.: Some theorems on saturated homomorphs of soluble Lie algebras. Math. Z.115, 179-187 (1970) · Zbl 0197.03003 · doi:10.1007/BF01109856
[4] Barnes, D.W.: Sortability of representations of Lie algebras. J. Algebra to appear · Zbl 0272.17002
[5] Jacobson, N.: Lie Algebras. New York-London: Interscience 1962 · Zbl 0121.27504
[6] Tuck, W.: The Frattini subalgebra of a Lie algebra. Ph.D. Thesis, University of Sydney 1969 · Zbl 0169.28301
[7] Zassenhaus, H.: On trace bilinear forms on Lie-algebras. Proc. Glasgow math. Assoc.4, 62-72 (1959) · Zbl 0135.07303 · doi:10.1017/S204061850003389X
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