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| Automorphisms of the Projective Quaternion Unimodular Group of twoDimensions |
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Citation： |
Wan Zhexian,Yang Jingen.Automorphisms of the Projective Quaternion Unimodular Group of twoDimensions[J].Chinese Annals of Mathematics B,1982,3(3):395~402 |
| Page view： 892
Net amount： 1185 |
Authors： |
Wan Zhexian; Yang Jingen |
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| Abstract： |
Let K be the skew field of rational quaternoions.Let R={(a+bi+cj+dk)/2|a,b,c,d =in Z and have the same parity},where Z denotes the ring of rational integers.R is a subring of K and K is the quotient skew field of R. R is usually called the ring of quaternion integers.
Let E denote the subgroup of GL_2(R) generated by all elements of the form $[\left( {\begin{array}{*{20}{c}}
1&s\0&1
\end{array}} \right)\]$ and $[\left( {\begin{array}{*{20}{c}}
1&0\t&1
\end{array}} \right)\]$(s,t \in R).Denote the factor groups of GL_2(R) and E modules their centers,both of which are {\pm I},by PGL_2(R) and PE respectively.PE is the commutator subgroup of PGL_2(r).
Theorem.Any automorphism of PGL_2(R) (or PE) is one of the following two standard forms
$\bar A \mapsto \bar P{\bar A^\sigma }{\bar P^{ - 1}}$
$[A \mapsto \bar P{(\overline {{A^{\tau '}}} )^{ - 1}}{\bar P^{ - 1}}$
where $\bar P \in PGL_2(R)$,\sigma is an automorphism of R and \tau is an anti-automorphism of R. |
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