1. Let G and H be groups, and let g E G and he H. Find a formula for the order of the element (g, h) E G x H in terms of the order of g in G and the order of h in H.

Determine which of the frieze groups is the symmetry group ofeach of the following patterns.a. . . . D D D D . . .b. . . .V V V V . . .c. . . . L L L L . . .d. . . . V V V V . . .e. . . . N N N N . . .f. . . . H H H H . . .g. . . . L L L L . . .

he following is a multiplication table for a group G. The product a * b is given in the ble where a is from the leftmost column and b from the top row. Let H e V W X Y 2 = e e V W X W V Y Z X W X Y e W X Z V SOSN e What are the orders of the elements? X 3 Y 2 Y 2 Z X V Y e e X W V W e V Y 3 NS Y ১৯৯ {e, z). Show that this is a subgroup of G. Calculate the right cosets of H. Without calculating the left cosets of H, explain why H is a normal subgroup of G. Construct a multiplication table for the quotient group G/H. Is G/H cyclic? Give reasons. Describe the subgroup Z ≤ G/H generated by the element Hx = G/H. What is the order of (G/H)/Z?

. Let φ : G → G0 be a homomorphism of groups, and let H be a subgroup of G.