1) Suppose that a is a group element and a raised to power 6 equals e. What are the possibilities for |a|? Provide justification for your answer.

2) Consider the set {4,8,12,16}. Show that this set is a group under multiplication modulo 20 by constructing its Cayley table and providing an explanation. What is the identity element? Is the group cyclic? If so, provide all of its generators.

3) Find Aut(Z6).

4) Prove that a group of order 63 must have an element of order 3.

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