theorem A.338 Engel

open in the book · appendices/A-long-proofs.tex:16420 · p. 2955

Rests on

depends_on definition A.331 Derived series, solvable, ideal ¶
depends_on lemma A.337 A nilpotent map has nilpotent adjoint ¶
- depends_on definition 5.37 Linear transformation ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.32 Field ¶
      - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.29 Action ¶
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - depends_on definition 4.30 Ring ¶ ↺
- proves proof app:A-long-proofs@proof-205 ¶
proves proof app:A-long-proofs@proof-206 ¶

Supports

depends_on theorem A.340 Cartan's criterion for solvability ¶
- depends_on corollary A.342 Orthogonal splitting of ideals ¶
  - depends_on corollary A.343 The Killing form of a simple ideal ¶
    - depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶
      - depends_on theorem A.410 Whitehead's first and second lemmas ¶
  - depends_on lemma A.406 Splitting a submodule of codimension one ¶
    - depends_on theorem A.407 Weyl's complete reducibility theorem ¶
      - depends_on proposition A.408 The two groups vanish for the trivial module ¶
      - depends_on theorem A.410 Whitehead's first and second lemmas ¶ ↺
  - depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶ ↺
  - depends_on proposition A.408 The two groups vanish for the trivial module ¶ ↺
  - depends_on theorem A.407 Weyl's complete reducibility theorem ¶ ↺
- depends_on lemma A.402 The trace form is nondegenerate for a faithful representation ¶
  - depends_on lemma A.406 Splitting a submodule of codimension one ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Derived series, solvable, ideal	declared	appendices/A-long-proofs.tex:16427
`depends_on`	→	A nilpotent map has nilpotent adjoint	declared	appendices/A-long-proofs.tex:16427
`depends_on`	←	Cartan's criterion for solvability	declared	appendices/A-long-proofs.tex:16572
`proves`	←	app:A-long-proofs@proof-206	declared	appendices/A-long-proofs.tex:16430