The physical distinguishing feature for the special subgroups of the matrix groups $U(n)$, $O(n)$ etc seems to be that the handedness of the coordinate system does not change if the transformation belongs to a special subgroup. I have thought about it but cannot make any connection or headway to prove it. How would I go about characterizing this handedness to show that it is left unchanged under such transformations?
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The answer to your question depends on your understanding of "handedness".
The easy "abstract nonsense" answer goes as follows: Two bases $(e_1,\ldots,e_n)$ of the given $n$-dimensional real vector space $V$ are called equivalent if the unique linear transformation $T:\ V\to V$ defined by $Te_i:= f_i$ $(1\leq i\leq n)$ has positive determinant. It is easy to see that there are exactly two equivalence classes. One says that two bases in the same class have the same handedness. Since transformations $T\in SO(n)$ have determinant $1$ they preserve the handedness of any system of $n$ linearly independent vectors $x_1$, $\ldots$, $x_n$.
But of course you desire a geometrical intuitive or "physical" explanation of this equivalence relation. Now this is a more complicated story. The set of all bases $(e_1,\ldots,e_n)$ of $V$ is an $n^2$-dimensional space $X$ in its own right, and it turns out that $X$ consists of exactly two disjoint components. Two bases in the same component can be continuously deformed into each other so that at each time $t$ between $0$ and $1$ we have $n$ linearly independent vectors. (The proof of this fact is too long for this answer.) This means that we have a continuous map $t\mapsto T_t\in GL(n)$ with $T_0(e_i)=e_i$ and $T_1(e_i)=f_i$ $(1\leq i\leq n)$.
But for two bases not in the same component such a deformation is not possible. To prove the latter statement one has to look at $\det(T_t)$ as a function of $t$. This determinant is $=1$ at $t=0$ and $<0$ at $t=1$, so it has to vanish for some $t$ in between.
Those transformations that preserve the handedness are simply those with a positive determinant. One can easily see if two transformations have a positive determinant so does their product and inverses, hence they form a subgroup.