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- If
*A*and*B*are two subgroups of*I*(**R**^{2}) each generated by a rotation by 2π/*n*(about different points of**R**^{2}), then prove that*A*and*B*are conjugate subgroups.

(i.e. for some*g*∈*I*(**R**^{2}) we have*A*=*g*^{-1}*Bg*.)

Deduce that although there are many different subgroups of*I*(**R**^{2}) isomorphic to*C*_{n}they are all conjugate.Prove the same result for subgroups isomorphic to

*D*_{n}.Prove that although the subgroups

*D*_{1}and*C*_{2}are isomorphic as groups, they are not conjugate subgroups of*I*(**R**^{2}). - A subgroup of symmetries of a set
*S*is called*discrete*if there is some distance*m*such that every element of*S*is either fixed or moved by at least*m*by every element of*G*.

If*G*is a discrete subgroup of*I*(**R**), prove that*G*contains a translation*T*by a minimum distance and that every translation in*G*is a power of*T*. - Prove that there are two infinite discrete subgroups of the group
*I*(**R**), one generated by a translation (which we will call*C*_{∞}**Z**under addition) and the other generated by a pair of reflections (which we will call*D*_{∞}).Prove that the

*Frieze groups*(i), ... , (vii) considered earlier are (respectively) isomorphic to*C*_{∞},*C*_{∞},*D*_{∞},*D*_{∞},*D*_{∞},*C*_{∞}×*D*_{1},*D*_{∞}×*D*_{1}. *Inversion in a point*∈**a****R**^{3}or*reflection in*is the map**a****x**2**a**-**x**.

A*rotatory inversion*is rotation about a line*L*followed by inversion in a point of*L*.

Show that every rotatory reflection is a rotatory inversion and vice versa.- Define a map
*θ*from*O*(3) to the direct product*SO*(3) × {±1} by

*θ*(*A*) = ( (*det*(*A*).*A*,*det*(*A*) ).

i.e.*θ*(*A*) = (*A*, 1 ) if*A*∈*SO*(3) and ( -*A*, -1 ) if*A*∉*SO*(3).

Prove that*θ*is a group isomorphism. - If
*X*is a subset of**R**^{3}with -*X*=*X*(i.e.**x**∈*X*if and only if -**x**∈*X*) prove that the group*S*(*X*) of all symmetries of*X*is isomorphic to*S*_{d}(*X*) × <*J*> where*S*_{d}(*X*) is the subgroup of direct symmetries of*X*and <*J*> is the group of order 2 generated by*J*:**x**-**x**.

Solution to question 5

- Define a map
- Three identical cards with dimensions 2 × 2
*a*for*a*a real number are assembled as shown with their centre lines along the axes. Write down the coordinates of the twelve corners of the cards.

Show that if the number*a*is the Golden ratio*φ*=^{1}/_{2}(1+√5) then the distance*AB*is equal to the distance*AC*and deduce that the twelve points are at the vertices of an icosahedron.

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