# Dihedral notation dilemma

There seem to be two notations for a dihedral group of order $2n$.
$$D_n\quad and \quad D_{2n}$$
Both have logical rationales, the underlying cycle (or polygon) for one and the order of the group for the other. The Wikipedia article on dihedral groups claims the first is used in geometric contexts and the second in algebraic contexts. I didn't research the source for that claim. Alternative notations are not uncommon, but in this case they are very close, make a significant difference, and can be hard to discern from context unless there is a lot of context.

I have reviewed the last 30 posts tagged "dihedral groups." Only two of them used $D_{2n}$. One of these two posts quoted a text by Dummit and Foote that used $D_{2n}$. For reference this is the other post. The response to a third post took the first few lines to explain that $D_n$ and not $D_{2n}$ was going to be in the answer.

I am not familiar with current undergraduate texts in Algebra or Group Theory, so I don't know which notation is currently popular among textbook writers. However, the use of $D_n$ seems to be popular among our users.

Philosophers tell us it is unwise to derive an ought from an is. In any case, we would not want to impose a standard on people posting questions. However, a nudge toward standardization might be helpful. With this in mind, I added a suggestion to the description of the tag "dihedral group" that reads as follows: "The group is usually denoted $D_n$ if the polygon has $n$ sides." (A polygon was mentioned in the prior sentence.) This addition is under review and may or may not be accepted in accordance with the criteria for tags. In any case, I would like to see this topic discussed.

1. Is $D_n$ the better choice of the two?
2. Is using the description of the tag a helpful nudge? Are there any other nudges that would work better or in addition?
3. Is this a good or bad use of the tag description? (There may already be a policy on this.)
4. Are there other notation standards that might be good to encourage?
• Related tag edit results here and here
– user99914
Nov 25 '17 at 6:32
• I was not aware of the difference in notations while reviewing your edit. I have essentially undone your edit.
– user99914
Nov 25 '17 at 6:38
• Thanks for giving me another reason never to buy Dummit & Foote. The earlier deal breaker was their heretic view of honoring structures without a multiplicative identity with the status of a ring. Nov 25 '17 at 10:30
• Anyway, while I agree that $D_n$ is better, this is still just a convention. Of course, if there is a lone wolf arguing for a different convention, we should just ignore them and burn their produce in a bonfire (a fitting fate for the drivel promoting $\tau$ over $2\pi$). My annoyance comes largely from the need to add things to the list of warnings I need to give to my students. Nov 25 '17 at 10:34
• IMO the overriding principles in subscript parametrization are A) the parameter should be natural, and B) efficiently fill the range of values. A case can be made both ways re item A, but item B seems forgotten here. If they want to make it easier for the students to remember the order of the group then why don't they do the same with the symmetric groups also. Surely $S_{24}$ and $S_{40320}$ are then to be preferred over $S_4$ and $S_8$. Nov 25 '17 at 10:39
• Ok, enough of me blowing off esteem. If there is more than a single tome promoting both sides, I guess we need to be accomodating and warn the readers/students about the differences of opinion/notation. So the tag wiki should mention both. Then we have done our duty of informing the users. Nov 25 '17 at 10:44
• Related to one of @JyrkiLahtonen's comments: Does anyone believe that there are rings without unit elements? Nov 25 '17 at 12:17
• Wikipedia seems to prefer $D_{2n}$ as seen here and on similar pages. As a recent university graduate, I can confirm that (at least at my alma maters) $D_{2n}$ was preferred in many contexts in both mathematics and computer science. Also, my undergraduate group theory course used D&F as a reference and used the same notation found therein. Nov 25 '17 at 19:44
• Use whichever notation you find more suitable for context... but be sure when you introduce the $D_n$ to specify its order as $n$ or $2n$. Nov 26 '17 at 2:57
• @Stella Biderman Thanks for your comment. I checked out the Wikipedia article on dihedral groups in general; it uses the convention that the order of $D_n$ is $2n$ as does Wolfram MathWorld and they both have references. The Wikipedia articles on specific $D_n$s such as $D_8$ seem to come from a subwiki called Groupprops for which no references are provided. Nov 26 '17 at 4:00
• @CogitoErgoCogitoSum The laissez faire approach you suggest is certainly preferable in general. However, ambiguous notation can create problems when used by the OP. It requires questions back to the OP, losing time and potentially causing loss of interest in both parties. My point is that a single unambiguous convention should be encouraged, I have a preference, but it pales in comparison to having a quasi-standard. Nov 26 '17 at 4:35
• We should just all agree to call it $I_2(n)$ and avoid the confusion. Nov 26 '17 at 14:49
• @Tobias what's the reference or the joke? Nov 26 '17 at 15:18
• The notation I gave is completely unambiguous, it is just only used in certain contexts (namely the context of Coxeter groups, plus it requires the understanding that the notation refers to the group corresponding to the given type). Nov 26 '17 at 15:21
• @TobiasKildetoft so it wasn't really serious but it was more serious than I-tu-n(es). Nov 26 '17 at 19:32