Automorphisms of the Projective Quaternion Unimodular Group of twoDimensions

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 $[left( {begin{array}{*{20}{c}}1&s0&1end{array}} right)]$ and $[left( {begin{array}{*{20}{c}}1&0t&1end{array}} right)]$(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.