What notation is most common for the dihedral group of order $2n$? I'm talking about the group of symmetries of a regular $n$-gon. I know that some books call this group $D_n$, and some books call it $D_{2n}$. There are probably other notations as well.
What notation, if any, is considered more standard? Is there a difference (as there often is) between the US and Europe?
I know that, if I'm writing anything about such a group, I will need to define my notation in context; I'm not asking so that I can be lazy and just use one notation without specifying what it means. I mention this only to prevent people lecturing me about defining my terms in mathematical writing - you would be preaching to the choir. I just wonder what, if anything, people view as "standard".
Thanks in advance.
In my experience, the notation $D_{2n}$ is mostly used by group theorists while the notation $D_n$ is mostly used by everyone else.
What is the reason group theorists think it's "consistent" to write dihedral groups as $D_{2n}$ but they write symmetric groups as $S_n$ like everyone else instead of $S_{n!}$? I once asked Richard Foote about this ("Would you call the permutations of $5$ letters $S_{120}$?") and here was his answer: in practice, if an abstract symmetric group arises you can nearly always find a set of $n$ things that it permutes, but abstract dihedral groups usually arise with no regular $n$-gon involved in some way, so group theorists prefer to label dihedral groups by their size, just as we all do for the quaternion group $Q_8$ (and $Q_{2^n}$, or even more generally $Q_{4m}$). I thought that was a nice explanation, which I use to this day when telling students in a course why I'll write $D_n$ in lectures and course assignments or tests even if the book for the course uses $D_{2n}$.