# I want examples of definable groups in algebraically closed fields?

I can not think of any example of a definable group in algebraically closed fields?

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When you say definable, can you specify the language and theory? I assume that you mean language of rings and the theory is ACF, but the more you write the better answers you might get. – Asaf Karagila Mar 21 '12 at 9:12

Algebraic groups are the most relevant examples that are definable over alegebraically closed fields. So you can take $\mathrm{GL}(\mathfrak{n},\mathbb{C})$ can serve as an example.
Its nothing but the General Linear Group of set of invertible matrices over $\mathbb{C}$. So if you are looking for some more examples see this where the author gives numerous examples of such groups in $8.1$ @sagara – Iyengar Mar 21 '12 at 9:24
Ummm, Iyengar is right. "The set of invertible matrices" easily unpacks to the first order definition $\{ (x_{11}, \ldots, x_{nn}) : \exists (y_{11}, \ldots, y_{nn}) \mbox{ such that } (x_{ij}) (y_{ij}) = \mathrm{Id}_n \}$. Here the matrix identity must be written out as $n^2$ explicit equations of the form $\sum_k x_{ik} y_{kj} = \delta_{ij}$. (There are more efficient ways to define $GL_n$, using determinants, but this is the most straightforward interpretation of what Iyengar wrote.) – David Speyer Mar 21 '12 at 14:07