Isogeny in superstable groups

Freitag, James

doi:10.1007/s00153-014-0373-z

Cited by 1 publication

(6 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The strongly connected component is a definable, normal, and strongly connected group. For more details of this sort, see [Fre14].…”

Section: Almost Simple Superstable Groupsmentioning

confidence: 99%

“…Inspired by [CS11], in [Fre14] it was shown that any superstable group admits a subnormal decomposition series so that the quotients are almost simple superstable. Isogeny is an equivalence relation on stongly connected superstable groups.…”

Section: Almost Simple Superstable Groupsmentioning

confidence: 99%

“…In light of this result, it seems natural to ask what additional properties one might prove regarding the structure of the almost simple superstable groups. When G is almost simple superstable, G/Z(G) is always simple or abelian (see [Fre14] for a proof). In this paper we will be considering the case in which G/Z(G) is simple.…”

Section: Almost Simple Superstable Groupsmentioning

confidence: 99%

“…The infinite simple quotients in the respective series are, by the theorem (after some appropriate permutation of the indices), isogenous as almost simple superstable groups. However, the kernel of any isogeny of almost simple superstable groups is central, and thus trivial [Fre14]. Thus, the noncommutative simple groups which appear in the two series are (after some permutation) isomorphic as definable groups.…”

Section: Almost Simple Superstable Groupsmentioning

confidence: 99%

“…Following the proof of the theorem, we will explain interesting applications of the theorem in the differential algebraic context. Our proof uses the model theory of superstable groups coming from [BL86] and [Fre14] along with the theory of universal central extensions of Chevalley groups of [Ste67].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Superstability and central extensions of algebraic groups

Freitag

Minchenko

2016

Advances in Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Altinel and Cherlin proved that any perfect central extension of a simple algebraic group over an algebraically closed field which happens to be of finite Morley rank is actually a finite central extension and is itself an algebraic group. We will prove an infinite rank version of their result with an additional hypothesis, while giving an example which shows the necessity of this hypothesis. The inspiration for the work comes from differential algebra; namely, a differential algebraic version of the results here was used by the second author to answer a question of Cassidy and Singer. The work here also provides an alternate path to the same answer.An almost simple superstable group G is one in which every definable normal subgroup H has the property that RU(G) > RU(H) · n for any n ∈ N. We prove that any almost simple superstable group which is a central extension of a simple algebraic group is actually a finite central extension and is an algebraic group. We also explain the applications of this result to differential algebraic groups. Many of the central ideas of the proof of our main theorem are an adaptation of techniques developed by the second author in the setting of differential algebraic groups.

show abstract

“…The strongly connected component is a definable, normal, and strongly connected group. For more details of this sort, see [Fre14].…”

Section: Almost Simple Superstable Groupsmentioning

confidence: 99%

Section: Almost Simple Superstable Groupsmentioning

confidence: 99%

Section: Almost Simple Superstable Groupsmentioning

confidence: 99%

Section: Almost Simple Superstable Groupsmentioning

confidence: 99%