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Home / Modern Abstract Algebra / Dihedral group

Dihedral group

Dihedral group: A rotation of the plane lamina of a regular polygon which brings the vertices of the polygon into coincidance with themselves, is called a coincidence rotation or symmetry of the polygon. There are exactly 2n coincidance rotations of a regular polygon of n sides,

which form a group called the dihedral group, under the composition of coincidance rotations, and is denoted by n.

Problem: The group 3 of symmetries of an equilateral triangle.

Solution: ABC is an equilateral (a regular polygon of 3 sides).

Screenshot_4

Its counter clock wise rotations through 1200, 2400 and 3600 give three coincide rotations:

(A⟶B, B⟶C, C⟶A) = (A B C)

(A⟶C, B⟶A, C⟶B) = (A C B)

(A⟶A, B⟶B, C⟶C) = I, identity permutation I corresponds to no rotation or rotation 00.

Again the rotations of the plane lamina of triangle ABC about the altitude AD, BE, CF result in three coincidance rotations:

(A⟶A, B⟶C, C⟶B) = (BC)

(B⟶B, C⟶A, A⟶C) = (CA)

(C⟶C, A⟶B, B⟶A) = (AB)

Thus we have all the 6( = 2×3) coincidence rotations:

 

3 = {I, (AB), (BC), (CA), (ABC), (ACB)}

This is identical with the symmetric set S3 of all permutations on three symbols A, B, C.

3 is a group of order 6.

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