# Are symmetry axes in $D_n$ fixed?

I am studying the dihedral group and comparing the table I have in my book with that of wikipedia. I realize that the entries were inverted, that is XY in the wikipedia article corrisponds to YX in my book.

Let $$D_3=\{1,R1,R2,S1,S2,S3\}$$ be the dihedral group of the triangle- with $$R1$$ the counter-clockwise rotation of 120 degrees and $$R2$$ the the counter-clockwise rotation of 240 degrees. $$Si$$ is the reflection about the i-the symmetry axis(that is the axis from the vertex i to the midpoint of the opposite side) $$i=1,2,3$$.

I guess a possible explanation is that the definition of the symmetry axes is ambiguos:

If the axes of symmetry aren't fixed, they move with the vertices. What I mean is that for example: If I draw a triangle and number the vertices, initially the symmetry axis r2 corresponding to the vertex 2 has a certain position. In the operation $$S3S2$$ (first $$S3$$ and then $$S2$$): $$S3$$ changes the position of the vertex 2 and so does r2, then $$S2$$ will be the reflection with respect to the new position of the axis r2. So $$S3S2=R2$$.

If the axes are fixed geometrical objects,then in $$S3S2$$ then$$S3$$ changes the position of the vertex 2, but the axis labelled r2 is still the one at the beginnig, because it doesn't move. Then $$S2$$will be the reflection with respect to the initial position of the axis r2. This yields $$S3S2=R1$$.

So how should reflections be interpreted and what comes first in XY: X or Y?

• We can always relabel, and $XY$ is not uniquely defined. For me, as maps, it is $(XY)(i)=X(Y(i))$, so $Y$ first. Jun 22 '20 at 10:50