Quick definition question: Dihedral group

[srfriggen]

A dihedral group of an n-gon denoted by Dn, whose corresponding group is called the Dihedral group of order 2n?


What I gather from that is a square has 8 symmetries, an octagon has 16, a hexagon 12, etc?

 

[marschmellow]

Yep, you got it.

 

[LagrangeEuler]

When I do reflection in square do I need to put numbers and that do reflection for particular site of square. Is there some easiest way to find multiplication table of $D_4$?

 

[R136a1]

When I do reflection in square do I need to put numbers and that do reflection for particular site of square. Is there some easiest way to find multiplication table of $D_4$?

Let $R$ be a rotation and $S$ a reflection. Use the relations $S^2 = 1$, $R^4 = 1$ and $SR = R^3S$. Using this you can find the multiplication table easily.

 

[blue_raver22]

Yes, that is correct. The dihedral group of an n-gon is a group of symmetries that preserves the shape and orientation of the n-gon. This group has 2n elements, which correspond to the 2n possible rotations and reflections of the n-gon. Each element in the group represents a unique symmetry of the n-gon, and the group as a whole represents all possible symmetries of the n-gon. This concept is important in geometry, crystallography, and other areas of mathematics and science.