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I need an example involving the old invariant theory that pre-dated abstract algebra.

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Here is the example of Kleinian singularities. Let $G$ be a finite subgroup of $SU(2,{\bf C})$.

$G$ acts on ${\bf C}^2$ and for each $G$ corresponds a Klein singularity $$ S=\frac {{\bf C}^2 } {G} $$

$$ S=\{ (x,y,z) \in {\bf C}^3 \ | \ R(x,y,z)=0 \} $$

One example is the group ${\mathcal C}_{n}$, the cyclic group.

You can realize it as matrices of the form $$ \begin{pmatrix} \omega^j & 0 \cr 0 & \omega^{-j} \end{pmatrix} $$ where $\omega $ is a primitive root of unity. The invariant functions, are functions which satisfy $f(u,v)=f(au+bv, cu+dv)$

The following three functions are invariant $u^n$, $v^n$, $uv$. Set $x=u^n$, $y=v^n$ and $z=uv$. Then they satisfy the follwoing syzygy $$z^n=xy$$

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