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)

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 pro du t (5, c)(2, a). [1 mark℄(b) Find the inverse of (−3, b). [1 mark℄( ) Find the order |Z × V |. [1 mark℄(d) Is Z × V ab elian? (Justify your answer.) [1 mark℄(e) Find the order of (0, a) and the order of (1, b)

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

Find the orders of the following elements of the group C× = (C \ {0}, ×) of non-zero complexnumbers:(i) 1, (ii) −1, (iii) 3, (iv) i

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.

8. (a) Let G b e the group Z2 ⊕ Z2. Show that G is not y li . [5 marks℄(b) Let G b e the group Z3 ⊕ Z2. Show that G is y li . [4 marks℄