Asking thisthis question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now I am completely lost =)
My original question goes as follows (with a little editing).
Q1: Suppose we have a linear operator $T:\mathbb{R}^n\to \mathbb{R}^n$, $n>2$, which is NOT a constant times an isometry. Let $G$ be the subgroup of $GL_n(\mathbb{R})$, generated by the orthogonal group together with $T$. I need to prove (or refute) that $G$ acts "bitransitively" on $\mathbb{R}^n$, in the sense that it can map any pair of non-colinear vectors into any other pair of non-colinear vectors. I also need a similar result for the complex case.
A comment that I received states that this is true, due the following:
Q2: $\mathbb{R}^*O_n(\mathbb{R})$ is a maximal subgroup in $GL_n(\mathbb{R})$ (and analogously for the complex case).
I don't really see neither why these statements hold, nor how to derive my question from them, except of the case when $\det T=\pm1$. I have found two papers that seem to have proofs of these statements, but I don't really have a background in algebra, so I can't even understand if this is the case (I am not sure if the Euclidean spaces fit into the scope of these papers). Furthermore, I am convinced that there should be some geometry behind this issue, and there is no chance that I can retrieve the geometric intuition from theses papers, again since my background doesn't allow me to read them. And as I've mentioned before, I don't see how to derive Q1 from Q2 in general case.
I hope you can help me with this predicament.
I will also add another related (vaguely stated) question.
Q3: What can be said about a subgroup $G$ of $GL_n(\mathbb{R})$, if it can map any $k$-tuple of linearly independent vectors into any other $k$-tuple, for some $k\in\overline{1,n-1}$?
