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Can one give an elementary proof of following interesting theorem? (These are from the paper "Finite subgroups of $PGL(2,K)$-Beauville").

(In each statement , characteristic of $K$ is prime to order of group in statement, as mentioned in the paper above. One may modify the statement if possible. )

  1. $PGL(2,K)$ contains $\mathbb{Z}/r$ and $D_r$ iff $K$ contains $\zeta +\zeta^{-1}$ for some premitive $r^{th}$ root of unity $\zeta$.

  2. $PGL(2,K)$ contains $A_4$ and $S_4$ iff $-$ is sum of two squares in $K$.

  3. $PGL(2,K)$ contains $A_5$ iff $-1$ is sum of squares in $K$ and $5$ is a square in $K$

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I think these are all just statements about the fields of definition of the faithful two-dimensional representations of the respective group. –  Alex B. Aug 30 '11 at 4:13
    
@Alex: The arguments given in the paper mentioned above involve the same things you said; but I couldn't understand. So I have written "...elementary proof..." –  user8186 Aug 30 '11 at 4:58
    
If you are interested in learning the necessary background material, see e.g. my notes on representation theory: math.postech.ac.kr/~bartel/docs/reptheory.pdf or any of the standard books on the topic. –  Alex B. Aug 30 '11 at 7:46

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