Journal of Prime Research in Mathematics

# Elementary calculus in chevalley groups over rings

Alexei Stepanov
Department of Mathematics and Mechanics, Sankt-Petersburg State University, Russia.
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan.
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$$^{1}$$Corresponding Author: stepanov239@gmail.com

### Abstract

The article studies structure theory of Chevalley groups over commutative rings. Main results of the article are relative dilation and local-global principles. and an economic set of generators of relative elementary subgroup. These statements proved by computations with elementary unipotents (hence the title) are very important in further development of the subject. No restrictions on the ground ring or the root system $$Φ$$ are imposed except that the rank of $$Φ$$ is not less than 2. The results improve previous results in the area. The article contains a brief survey of the subject, some gaps in proofs or incorrect references are discussed. Proofs of some known related results are substantially simplified.

#### Keywords:

Chevalley groups, principal congruence subgroup, local global principle, dilation principle.