ind the order of each element of the dihedral group D4 of symmetries of a square

The dihedral group G = D8 has elements {ρ0, ρ90, ρ180, ρ270, µH , µV , µD, µU }.Its multiplication table is partially completed below.ρ0 ρ90 ρ180 ρ270 µH µV µD µUρ0 ρ0 ρ90 ρ180 ρ270 µH µV µD µUρ90 ρ90 ρ180 ρ270 ρ0 µU µV µHρ180 ρ180 ρ270 ρ0 ρ90 µV µH µU µDρ270 ρ270 ρ0 ρ90 ρ180 µU µD µH µVµH µH µU µV µD ρ0 ρ180µV µD µU ρ180 ρ0 ρ90µD µD µH µU µV ρ90 ρ270 ρ0 ρ180µU µU µV µD µH ρ270 ρ90 ρ180 ρ0(a) Calculate the missing entries. (You are only required to write down themissing entries, you do not need to copy out the table.) [3 marks](b) Give an example of a cyclic subgroup of G and an example of a non-cyclicsubgroup of G. [3 marks](c) Show that K = {ρ0, ρ180} is a subgroup of G. [2 marks](d) Find all the right cosets of K = {ρ0, ρ180} in G.

Consider the direct product D8 × S4 where D8 is the group in Question 6 andS4 is the symmetric group (with its usual operation).(a) Find the product(µD,(1 2 3 42 4 3 1)) (ρ90,(1 2 3 41 4 2 3)).[2 marks](b) Find the inverse and the order of(ρ180,(1 2 3 42 4 3 1)).[4 marks]8. (a) State Lagrange's theorem and use it to show that if G is a group of orderp2, where p is prime, and H is a non-cyclic subgroup of G then G = H.[6 marks](b) Let H be a subgroup of a group G and let a, b ∈ G be such that aH ∩bH 6 =∅. Show that aH ⊆ bH. [4 marks]4

Quadrilaterals must have exactly 4 sides.The shapes in must have 2 pairs of parallel sides, 4 right angles, and 4 sides of equal length.The shapes in must have at least one pair of parallel sides.The shapes in Group D must have 2 pairs of parallel sides and 4 sides of equal length.The shapes in must have 2 pairs of parallel sides and 4 right angles.The shapes in must have exactly 2 pairs of parallel sides.

Consider the dire t pro du t Z × V where Z is the group of integers (under ad-dition) and V is the Klein 4-group with elements {e, a, b, c} and multipli ation∗ dened by the table b elow:* e a b ce e a b ca a e c bb b c e ac c b a e(a) Find the order |Z × V |. [1 mark℄(b) Is Z × V ab elian? (Justify your answer.) [1 mark℄( ) Find the order of (0, a) and the order of (1, b)

Which geometries are possible when there are four electron groups around the central atom?