Signalizers and balance in groups of finite Morley rank

Burdges, Jeffrey

doi:10.1016/j.jalgebra.2008.12.001

Cited by 7 publications

(12 citation statements)

References 28 publications

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“…However, a more subtle definition is required to find an applicable version of this fact. The following definition was applied in [14], under the assumption of Prüfer rank at least 3. Definition 1.10.…”

Section: Components and Descentmentioning

confidence: 99%

“…We recall from [14] that theB-property holds in the absence of a proper 2-generated core. The proof of this result explains the definition ofr * .…”

Section: Components and Descentmentioning

confidence: 99%

“…In high Prüfer rank, Burdges [14] has produced an algebraic group. In other situations, we will reach a contradiction, as shown below.…”

Section: Proof Of the Trichotomy Theoremmentioning

confidence: 99%

“…The present paper lies between the high Prüfer rank, or generic, where general methods are used heavily, and the 'end game', where general methods give way to consideration of special cases. In the first part, the generic trichotomy theorem [14] says that a minimal nonalgebraic simple group of finite Morley rank has Prüfer rank at most 2. Thus we may consider small groups with simple sections restricted to PSp 4 , G 2 , PSL 3 , and PSL 2 .…”

Section: Introductionmentioning

confidence: 99%

“…For traditional and technical reasons, we prefer to record this result as a tameness-free version of the old trichotomy theorem from [7], using the generic trichotomy theorem [14] (page 8) when the Prüfer 2-rank is at least 3.…”

Section: Introductionmentioning

confidence: 99%

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A new trichotomy theorem for groups of finite Morley rank

Borovik¹,

Burdges²

2007

Journal of the London Mathematical Society

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There is a longstanding conjecture, of Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. Here we will conclude that a simple K * -group of finite Morley rank and odd type either has normal rank of at most 2, or else is an algebraic group over an algebraically closed field of characteristic not 2. To this end, it suffices to produce a proper 2-generated core in groups with Prüfer rank 2 and normal rank at least 3, which is what is proved here. Our final conclusion constrains the Sylow 2-subgroups available to a minimal counterexample and, finally, proves the trichotomy theorem in the nontame context.

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Section: Components and Descentmentioning

confidence: 99%

“…We recall from [14] that theB-property holds in the absence of a proper 2-generated core. The proof of this result explains the definition ofr * .…”

Section: Components and Descentmentioning

confidence: 99%

“…In high Prüfer rank, Burdges [14] has produced an algebraic group. In other situations, we will reach a contradiction, as shown below.…”

Section: Proof Of the Trichotomy Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations