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ry22
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I don't understand how to show that the reflections and rotations are associative. Thanks for any help.
ry22 said:I don't understand how to show that the reflections and rotations are associative. Thanks for any help.
A dihedral group Dn is a group of symmetries of a regular n-gon. It is denoted by Dn and has 2n elements.
To prove closure, we need to show that the composition of any two elements in Dn results in another element in Dn. This can be done by showing that the composition of any two rotations or reflections in Dn is also a rotation or reflection in Dn.
The identity element in Dn is the rotation by 0 degrees, which leaves all points in the n-gon unchanged.
To prove the existence of inverses, we need to show that for every element in Dn, there exists another element in Dn that when composed together, results in the identity element. In Dn, the inverse of a rotation is the rotation in the opposite direction and the inverse of a reflection is itself.
The order of Dn is 2n, as it has 2n elements. This can be seen by considering the number of rotations and reflections in Dn.