3. Let G = Z4xZ7. a. Determine all the generators for G. b. Determine a subgroup for each order 14, 7, 4, and 2. Then draw a subgroup lattice diagram of G. c. Let N = ((0, 1)). Determine the right cosets of N.

Let H be a subgroup of G and let a, be G. If Ha = Hb, then * aH = b-H a-1H = b-1H Ha = Hb Ha-1 = Hb-1 Ha-1 = Hb-1 None of the choices

Consider the group D4 and the subgroup {I, F}. List all the left cosets of H (with all theelements in each), indicating which are equal. Then, list all the right cosets of H.

1. Draw the group tables for the following groups:  (ii) The group D4 of symmetries of a square is given by D4 = {I, R, R^2, R^3, H, V, D, D0} where I is the identity, R is a clockwise rotation through π/2, H and V are reflections in the horizontal and vertical, and D, D0 are reflections in the diagonals y = x and y = −x respectively.