Exercise 5.1. Let A be a set. A subgroup H of SA is transitive on A if for each a, b = A, there exists σ = H such that σ(a) = b. (1) Show that if H is transitive on A, then |H|≥ |A|. (2) Show that if A is nonempty and finite, then there exists a subgroup H of SA with ||H| = |A| that is transitive on A. (3) Suppose A is nonempty and finite, and H is a subgroup of S¸ with |H| that is transitive on A. Prove or disprove that H is cyclic. = |A|

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