1970
DOI: 10.2140/pjm.1970.34.177
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Frattini subalgebras of a class of solvable Lie algebras

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Cited by 18 publications
(23 citation statements)
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“…Since L 2 is abelian, the Fitting null component of ad(x) acting on H is nilpotent; hence it is a Cartan subalgebra of H. Hence we can let D be the Fitting null component. Since φ(H) = 0, Proposition 1 of [7] yields that D, hence ad(x), acts completely reducibly on H since x ∈ D, H = D + H 2 and D is abelian. Since K is algebraically closed, ad(x) acts diagonally on H and on the ideals L 2 and φ(L) that are contained in H. These results hold for all x ∈ C and, since C is abelian, the ad(x) are simultaneously diagonalizable on L 2 and on φ(L) .…”
Section: Lemma Let L Be a Minimal Non-elementary Finite Dimensional mentioning
confidence: 99%
“…Since L 2 is abelian, the Fitting null component of ad(x) acting on H is nilpotent; hence it is a Cartan subalgebra of H. Hence we can let D be the Fitting null component. Since φ(H) = 0, Proposition 1 of [7] yields that D, hence ad(x), acts completely reducibly on H since x ∈ D, H = D + H 2 and D is abelian. Since K is algebraically closed, ad(x) acts diagonally on H and on the ideals L 2 and φ(L) that are contained in H. These results hold for all x ∈ C and, since C is abelian, the ad(x) are simultaneously diagonalizable on L 2 and on φ(L) .…”
Section: Lemma Let L Be a Minimal Non-elementary Finite Dimensional mentioning
confidence: 99%
“…Then the following are equivalent: [3], and so L (1) is abelian, by Proposition 1 of [5]. Now (L m Y c Z{L) by Lemma 2.1, and so [7]).…”
Section: > P -Free Algebrasmentioning
confidence: 99%
“…In Sections 3,4 we seek analogues for </> p (L) of the results of Stitzinger on $(L) when the derived algebra L ( " is nilpotent, which were obtained in [5], The following notation will be used:…”
Section: Introductionmentioning
confidence: 99%
“…Stitzinger in [24] proved that a Lie algebra is an E-algebra if and only if L/φ(L) is elementary. He also proved that every strongly solvable Lie algebra over an arbitrary field is an E-algebra.…”
Section: The Solvable Casementioning
confidence: 99%
“…A Lie algebra L is called strongly solvable if L 2 is nilpotent. Stitizinger also proved in [24] that if L is strongly solvable then L is an E-algebra. In this paper it is shown that over a perfect field the converse also holds For algebraically closed fields of characteristic zero, elementary Lie algebras were determined by Towers in [27].…”
Section: Introductionmentioning
confidence: 98%