Isomorphisms and elementary equivalence of Chevalley groups over commutative rings

2019 ◽  
Vol 210 (8) ◽  
pp. 1067-1091 ◽  
Author(s):  
E. I. Bunina
Keyword(s):  
Commutative Rings ◽  
Chevalley Groups ◽  
Elementary Equivalence

Chevalley groups over commutative rings: I. Elementary calculations

1996 ◽  
Vol 45 (1) ◽  
pp. 73-113 ◽  
Author(s):  
Nikolai Vavilov ◽  
Eugene Plotkin
Keyword(s):  
Commutative Rings ◽  
Chevalley Groups

scholarly journals Elementary equivalence for finitely generated nilpotent groups and multilinear maps

1998 ◽  
Vol 58 (3) ◽  
pp. 479-493 ◽  
Author(s):  
Francis Oger
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Elementary Equivalence ◽  
Linear Map ◽  
Multilinear Maps

We show that two finitely generated finite-by-nilpotent groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. For each integer n ≥ 2, we prove the same result for the following classes of structures:(1) the (n + 2)-tuples (A1, …, An+1, f), where A1, …, An+1 are disjoint finitely generated Abelian groups and f: A1 × … × An → An+1 is a n-linear map;(2) the triples (A, B, f), where A, B are disjoint finitely generated Abelian groups and f: An → B is a n-linear map;(3) the pairs (A, f), where A is a finitely generated Abelian group and f: An → A is a n-linear map.In the proof, we use some properties of commutative rings associated to multilinear maps.

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