Question: Let $P_i\in\mathbb{R}^n$ be the hyperplane $x_i - x_{i+1} = 0$. Find a presentation for the group $G$ generated by the reflections in $P_1, \ldots, P_{n-1}$
Attempt: I really don't know how to start with this problem, which was introduced at the same time as group presentation notation. I understand that the determinant of a reflection through a hyperplane is $-1$, but then doesn't this mean that the composition of two such reflections would have determinant $1$, and hence not be a reflection? I would appreciate any hints as to how I could start this problem.
Here's a few hints.
Each $P_i$ is a linear transformation, and a linear transformation of $\mathbb R^n$ is determined by what it does to the standard basis $e_1,\ldots,e_n$. Furthermore, each $P_i$ permutes the set $\{e_1,...,e_n\}$, because it interchanges $e_i$ with $e_{i+1}$ and fixes all the other $e$'s. And then, since the list $e_1,...,e_n$ has no repetitions, you can even simplify the notation further and think of $P_i$ as a permutation of the index set $\{1,...,n\}$. This makes the group generated by the $P_i$'s isomorphic to a subgroup of the permutation group $S_n$ on the symbols $\{1,...,n\}$, where $P_i$ itself corresponds to the permutation that, written in cycle form, is $P_i = (i,i+1)$.
So if you know anything about the permutation group $S_n$, then you might be in a very good position to continue on with this problem......