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

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)

Which of the following groups of the periodic table contains both metals and non-metals?        I    Group IV       II    Group V      III    Group VIAI, II and IIIBII and III onlyCI and IIDI only

What is the determinant of an identity matrix of any order? a. The order of the matrix b. 1 c. Undefined d. 0

Define (i) group (ii) Ring (iii) Field