Dihedral Group D_4: Denotations & Correspondence

[Bashyboy]

In my class, we have denoted the elements of the dihedral group $D_4$ as $\{R_0, R_{90}, R_{180}, R_{270}, F_{\nearrow}, F_{\nwarrow}, F_{\leftrightarrow}, F_{\updownarrow} \}$. Not surprising, I was rather bewildered when I searched the internet for information on this group and encountered other denotations of the elements of $D_4$.

The standard denotation appears to be using $r$'s and $s$'s. I figured that $r = R_{90}$, $r^2 = R_{180}$, and $R_{270} = r^3$. However, I was unsure of the correspondence between the other elements. Could someone possibly help me with this?

 

[jbunniii]

Assuming $R_{90}$ denotes counterclockwise rotation by $90$ degrees, and we use the convention that $ab$ means "do $b$ first, then $a$", you can check that
$R_{90} F_{\updownarrow} = F_{\nwarrow}$
$R_{180} F_{\updownarrow} = F_{\leftrightarrow}$
$R_{270} F_{\updownarrow} = F_{\nearrow}$
so if we put $r = R_{90}$ and $f = F_{\nwarrow}$ then the group is generated by $r$ and $f$. Moreover, $r$ has order $4$ and $f$ has order $2$, and you can verify that $f r f = r^{-1}$; indeed, the relations $r^4 = f^2 = e$ and $frf = r^{-1}$ suffice to define the group.

 

[Bashyboy]

Okay, but what is the correspondence between $s$ and the flips (or reflections)?

 

[jbunniii]

Okay, but what is the correspondence between $s$ and the flips (or reflections)?

You can take $s$ to be anyone of the four flips/reflections, because all four of them satisfy $s^2 = e$ and $srs = r^{-1}$. Try it with a square sheet of paper to verify this.

To verify algebraically that it doesn't matter which flip we assign to $s$, first choose $s$ to be one of the flips and verify geometrically that it satisfies $s^2 = e$ and $srs = r^{-1}$. Then note that the other three flips are $rs$, $r^2s$, and $r^3 s$. Then if we set $f_k = r^k s$ for $k=0,1,2,3$, we compute
$$f_k^2 = f_k f_k = r^k s r^k s = r^k (s r^k s) = r^k r^{-k} = e$$
where the third inequality holds by repeatedly applying $s r s = r^{-1}$. So $f_k$ is its own inverse. Then
$$f_k r f_k = (r^k s) r (r^k s) = r^k (s r^{k+1} s) = r^k (r^{-(k+1)}) = r^{-1}$$
We have therefore shown that $f_k$ satisfies $f_k^2 = e$ and $f_k r f_k = r^{-1}$ for $k=0,1,2,3$, so any of the $f_k$ obeys the defining properties of $s$.

 

[blue_raver22]

I can provide some clarification on the denotations and correspondence of elements in the dihedral group $D_4$. The notation used in your class, with $R$ and $F$ denoting rotations and reflections respectively, is a common way to represent the elements of this group. However, as you have observed, there are other notations that are also used.

The standard notation for the dihedral group $D_4$ uses $r$ and $s$ to denote rotations and reflections respectively. In this notation, $r$ represents a rotation by $90$ degrees, $r^2$ represents a rotation by $180$ degrees, and $r^3$ represents a rotation by $270$ degrees. The correspondence between the elements in this notation and the one used in your class is as follows: $r = R_{90}$, $r^2 = R_{180}$, and $r^3 = R_{270}$.

The other elements in the group are reflections, denoted by $s$. In your class's notation, there are four reflections denoted by $F_{\nearrow}, F_{\nwarrow}, F_{\leftrightarrow}, F_{\updownarrow}$, which correspond to the four reflections in the standard notation: $s = F_{\leftrightarrow}$, $s^2 = F_{\updownarrow}$, $s^3 = F_{\nearrow}$, and $s^4 = F_{\nwarrow}$.

It is important to note that these notations are just different ways of representing the same group and its elements. The choice of notation depends on the context and the preference of the author. As long as the corresponding elements are clearly defined, both notations are valid and can be used interchangeably.

I hope this explanation helps clarify the denotations and correspondence of elements in the dihedral group $D_4$. If you have any further questions, please do not hesitate to ask. As scientists, it is important to have a clear understanding of the notations and terminology used in our field.