AUTOMORPHISMS OF SL (2, K) OVER SKEW FIELDS

In this paper, the author proves the following resu: It Let K be a skew field and A be an automorphism of SL(2, K). Then there exists A∈GL(2, K), an automorphism σ or an anti-automorphism τ of K, such that A is of theform AX=AX~σA~(-1) for all X∈SL(2, K)or AX=A(X~τ~2)~(-1)A~(-1) for all X∈SL(2, K),where X~σ, X~τ are the matrices obtained by applying σ, τ on X respee tively and X' is thetranspose of X.

AUTOMORPHISMS OF $\[SL(2,K)\]$ OVER SKEW FIELDS

In this paper, the author proves the following result Let $K$ be a skew field and $\\Lambda \]$ be an automorphism of $\SL(2,K)\]$. Then there exists $\A \in GL(2,K)\]$, an automorphism $\\sigma \]$ or an anti-automorphism $\\tau \]$ of $\K\]$, such that $\\Lambda \]$ is of the form $$\\Lambda X = A{X^\sigma }{A^{ - 1}}\]$$ for all $$\X \in SL(2,K)\]$$ or $$\\Lambda X = A{({X^{{\tau _1}}})^{ - 1}}{A^{ - 1}}\]$$ for all $$\X \in SL(2,K)\]$$ where $\{X^\sigma },{X^\tau }\]$ are the matrices obtained by applying $\\sigma \]$, $\\tau \]$ on X respectively and $\{X^'}\]$ is the transpose of X.