2
$\begingroup$

Is the image provided for a Cayley Diagram of $D_{4}$ in Wikimedia Commons wrong?

The image there seems to have the "outer ring" flipped:

enter image description here

Shouldn't this look more like this:

enter image description here

  • 0
    Your link points to the Cayley diagram of $D_4$, not $V_4$.2012-05-19
  • 0
    This arose in the context of [a question about using LaTeX to generate these diagrams](http://tex.stackexchange.com/q/56442/7844), to which readers here may wish to contributed answers.2012-05-19
  • 4
    In this diagram, the red arrow corresponds to multipication by $a$ *on the left*; your diagram is making it correspond to multiplication by $a$ *on the right*. So a red arrow that starts at $b$ should point to $ab$. Both conventions are okay, but it is true that the convention of the diagram clashes with the convention of the text. But the *description* in the text is accurate, as it identifies red arrows as "left multiplication by $a$".2012-05-19
  • 0
    @Mariano: It's the label that is wrong; the diagram posted here is that of $D_4$, and it is so identified in the Wiki page.2012-05-19
  • 0
    @Mariano: Yes! Typo: fixed.2012-05-19
  • 0
    @Arturo: I think I see. I'd (in my very limited experience) never seen the "left multiplication" convention used, and had always assumed that one (always) read the strings, from left to right, as corresponding to the order in which edges were traversed.2012-05-19
  • 0
    @Arturo: An an elaboration of your comment with a few words about which convention (if any) can be assumed in reading and drawing Cayley diagrams would be an answer.2012-05-19
  • 2
    @raxacoricofallapatorius: In Group Theory, you will often find that all lateral conventions occur; some people like left actions, some people like right actions. Some people have homomorphisms on the left, some on the right. Some people define commutators by $[x,y]=x^{-1}y^{-1}xy$, some by $[x,y]=xyx^{-1}y^{-1}$. Etc.2012-05-19
  • 0
    @raxacoricofallapatorius: If it helps to consider how multiplication on the left or right may make a difference,onsider the various objects involved (in particular, $a,b$) as real square $2\times 2$ matrices acting on vectors in the plane (represented as column or row vectors, as appropriate to the situation).2012-05-19
  • 0
    In particular, let $$a=\left(\begin{array}{cc} 0 & -1\\ 1 & 0\end{array}\right)$$ and $$b=\left(\begin{array}{cc} 1 & 0\\ 0 & -1\end{array}\right)$$.2012-05-19
  • 0
    @Arturo: What description are you citing that "identifies red arrows as 'left multiplication by a'"?2012-05-19
  • 1
    @raxacoricofallapatorius: In the [article on Cayley graphs](http://en.wikipedia.org/wiki/Cayley_graph). The definition (**Definition** section) agrees that an arrow labeled $s$ goes from $g$ to $gs$. But in the **Examples** section, the fourth example (which describes this particular graph), it explicitly states "Red arrows represent left-multiplication by element $a$."2012-05-19
  • 0
    @ArturoMagidin: Thanks, I see it now. (I was expecting it in the [description of the image in Commons](https://commons.wikimedia.org/wiki/File%3aCayley_Graph_of_Dihedral_Group_D4.svg#Summary).) Is there a general idiom for introducing the convention used in a Cayley diagram? In all the (admittedly elementary) examples I've seen, right multiplication is just _assumed_, without any explanation.2012-05-19
  • 1
    @raxacoricofallapatorius: Not that I am aware of; there really is no essential problem attached to this issue, since any argument that uses one convention can be done, *mutatis mutandis*, using the other convention. I think the article on Cayley graphs is a bit of a problem because it uses one definition and then has an example using the other convention, but I haven't taken the time to bring it up in the talk page (perhaps you can).2012-05-19

1 Answers 1

3

In the Wikipedia diagram, the red arrow corresponds to multipication by $a$ on the left; your diagram is making it correspond to multiplication by $a$ on the right. Both conventions are okay, but it is true that the convention of the diagram clashes with the convention of the text.

  • 0
    Adapted from comments above.2012-09-06