minimal module
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2010 ◽  
Vol 75 (2) ◽  
pp. 442-458 ◽  
Author(s):  
Ehud Hrushovski ◽  
James Loveys

AbstractWe give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.


2010 ◽  
Vol 32 (3) ◽  
pp. 533-538
Author(s):  
Qing-dong Huang ◽  
Lin-rang Zhang ◽  
Guo-min Li ◽  
Guang-yue Lu
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2007 ◽  
Vol 10 ◽  
pp. 329-340 ◽  
Author(s):  
A.J.E Ryba

We construct two embeddings of finite groups into groups of Lie type. These embeddings have the interesting property that the finite subgroup acts irreducibly on a minimal module for the group of Lie type. We present our constructions as examples of a general method that obtains embeddings into groups of Lie type.


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