About The Research To Structure Of Linear Group On Special Ring

This article theme is through to the research of special linear group, studies the linear group's structure on the special ring, simultaneously draws support from its subgroup's structure to inquire into its automorphic form, in the partial achievement foundation which the predecessor obtains, I absorb some domestic and foreign scholar's successful research mentality and the research technique, then has done the following research and innovation:1. Studied one kind of special linear group first: n orders cyclic matrix in assigns under the binary operation, including the ordinary matrix multiplication, Hadamard product and Fan product, respectively constitutes Abel group G1 , G2 and G3 , after the discussion I finally obtain the conclusions that these three Abel groups are the mutual isomorphism relations, namely G1 ? G2 ? G3.2. Studied the linear group automorphism on the exchangeable integral domain which characteristic number is not 2: if R is the exchangeable integral domain which characteristic number is not 2, when n≥3,δ( X )( X∈SL ( n, R)) is a isomorphic mapping from SL ( n, R ) to GL ( n, R ), then it must be one of the following two forms:δ( X )= P ?1XσP, ?X∈SLn ( R) orδ( X ) = P ?1 ( Xσ)' ?1P, ?X∈SLn ( R), in which matrix P satisfies the condition (5.1), andσis from R to its internal isomorphism, vice versa.3. Studied the linear group automorphism on the integral domains of principal ideal (not necessarily exchangeable) which characteristic number is not 2, and simplified proved the theorem that Wan Zhexian and Landin J and Riener I first obtained the linear group automorphism on the integral domains of principal ideal ring (not necessarily exchangeable) which characteristic number is not 2: if R is the integral domains of principal ideal (not necessarily exchangeable) which characteristic number is not 2, and n≥3,δ( X )( X∈SL ( n, R)) is a isomorphic mapping from SL ( n, R ) to GL ( n, R ), thenδ( X) must be one of the following two forms:δ( X )= P ?1XσP, ?X∈SL ( n, R)orδ( X )= P ?1 Xτ'?1P, ?X∈SL ( n, R), in which matrix P satisfies the condition (5.1), andσis from R to its internal isomorphism,τis from R to its internal anti-isomorphism, in which P∈GL ( n, R).4. Studied the linear group automorphism on Dedekind ring which characteristic number is not 2: if R is the Dedekind ring which characteristic number is not 2, when n≥3 the automorphism of GL ( n , R ) must be one of the following two forms:δ( X ) = P ?1μ( X )σi XσP orδ( X ) = P ?1μ( X )σi ( Xσ)' ?1P, in which matrix P satisfies the condition (5.1), andσis from R to its internal isomorphism,μis from GL ( n, R ) to R multiplication semi-group's homomorphism, vice versa.5. Through to the discussion and research of above results, and further refinement link's condition , can obtain the following precise result: if R is the exchangeable integral domain of identity element, which characteristic number is not 2, when n≥3 the automorphismδof HL ( n, R ) must be one of the following two forms:δ( X )= P ?1XσP, ?X∈HL ( n, R) orδ( X ) = P ?1 ( Xσ)' ?1P, ?X∈HL ( n, R), in which matrix P satisfies the condition (5.1), andσis from R to its internal isomorphism.This article research is helpful to know the theory of the finite group structure more deeply and more particularly, and innovates and develops the predecessor's partial achievements, to the linear group structure, and has certain significance to the linear group structure's research, specially to special link's linear group structure research.