Symmetry : Symmetries of a Square
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? Let S be a square, with vertices labelled (anticlockwise), 1,2,3,4. a symmetry of S is a rotation or reflection which preserves the square (although it may change the position of the vertices). Note that a symmetry is determined by its effect on the vertices.
a) List all 8 symmetries of the square.
b) Let a be a rotation about the centre of the square, anticlockwise through ( a quarter of a revolution). Let denote a rotation through in the clockwise direction. Let b be a reflection in the diagonal; line through vertices 1 and 3, and let e be the identity symmetry (which does nothing). Show that . (note that ba means do a then do b to the square, as we are writing maps on the left.)
c) Let G be the set of all 8 symmetries of the square. Show, using (a), that
(hint: write each of the symmetries you listed in (a) in the form )
d) you may assume without proof that G, together with composition of symmetries, is a group. It is called the dihedral group of order 8, and is denoted . Compute the subgroups H=<a> and K=<ba> of G.
e) compute the left cosets of H and the left cosets of K.
f) show that for and using that facts that
? in any group G,
? and that .
Show that . Your answer should not be a case by case proof!!!
g) deduce that H is a normal subgroup of G.
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Symmetries of a square are investigated. The solution is detailed, well presented and contains diagrams.
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