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℄

It seems like there might be some confusion in your question as the terms "y li" are not clear in the context of group theory. However, I can help you understand the groups Z2 ⊕ Z2 and Z3 ⊕ Z2.

(a) The group Z2 ⊕ Z2: This is the direct sum of two copies of the cyclic group of order 2. It has four elements: (0,0), (0,1), (1,0), and (1,1). The group operation is component-wise addition modulo 2. This group is not cyclic because there is no single element that can generate the entire group through repeated group operations.

(b) The group Z3 ⊕ Z2: This is the direct sum of the cyclic group of order 3 and the cyclic group of order 2. It has six elements: (0,0), (0,1), (1,0), (1,1), (2,0), and (2,1). The group operation is component-wise addition with the first component modulo 3 and the second component modulo 2. This group is also not cyclic because there is no single element that can generate the entire group through repeated group operations.

If "y li" refers to the group being cyclic, then neither group is cyclic. If "y li" refers to something else, please provide more context.