* 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).

Its counter clock wise rotations through 120^{0}, 240^{0} and 360^{0} 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 0^{0}.

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 S_{3} of all permutations on three symbols A, B, C.

∆_{3} is a group of order 6.