(8) Find cyclic subgroups of S 4 of orders 2, 3, and 4. (9) Find a subgroup of S 4 isomorphic to the Klein 4-group. List out its elements. (10) List out all elements in the subgroup of S 4 generated by (1 2 3) and (2 3). What familiar group is this isomorphic to? Can you ﬁnd four different subgroups of S 4 isomorphic to S 3?

QUESTION 2 (a) Suppose G is a cyclic group with generator g. Prove that every subgroup H of G is also cyclic. [11 marks] (b) Give a precise list (without repetition) of all the subgroups of Z108. Explain your answer. [5 marks] (c) Determine all the generators of the group 260. plain your answer. [9 marks]

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