Let $K$ be a field of characteristic 0, $n$ a positive integer, and $G$ the group $GL_n(K)$ of invertible linear transformations on $K^n$. Show that the following subsets of $G$ are $\emptyset$-definable: (a) the set of all scalar matrices; (b) the set of all matrices which are similar to a diagonal matrix with distinct scalars down the main diagonal; (c) the set of diagonalisable matrices.
My question is this. Are we viewing $G$ a model of the theory of groups and being asked to define subsets of $G$ in the language of groups, or are we viewing $K$ as a definable subset of $K^{n^2}$ and being asked to define subsets of $G$ in the language of fields? My first thought was the former, but I'm beginning to think the latter is intended.
I'm pretty sure it's the latter. This exercise is about linear algebraic groups, which are specific kind of algebraic groups, which in turn are definable groups in (algebraically closed) fields. If there are definable groups anywhere near this exercise, there's little room left for doubt.
I don't think the exercise is even doable in the other interpretation. It's easy to define scalar matrices (just the center), but I doubt you can define diagonal matrices using only group language in $GL_n$.