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By the Burnside Formula, number of orbits = 18 1 (6 + 1 + 3 + 4 + 3 + 1) = = 3. 6 6 So there are 3 orbits, namely {1, 2, 3}, {4, 5} and {6}. 40. A dinner party of seven people is to sit around a circular table with seven seats. How many distinguishable ways are there to do this if there is to be no ‘head of table’ ? 42 2. GROUPS AND GROUP ACTIONS Solution. View the seven places as numbered 1 to 7. There are 7! ways to arrange the diners in these places. Take X to be the set of all possible such arrangements, so |X| = 7!.

1-17. If A = [a0 ; a1 , . . , an ] with A > 1, show that 1/A = [0; a0 , a1 , . . , an ]. Let x > 1 be a real number. Show that the n-th convergent of the continued fraction representation of x agrees with the (n−1)-th convergent of the continued fraction representation of 1/x. 1 1 1-18. Find the continued fraction expansions of √ and √ . Determine as many conver5 5−1 gents as you can. √ 1 1-19. Investigate the continued fraction expansions of 6 and √ . Determine as many con6 vergents as you can.

There are two very different possible ‘sizes’ for sets, namely finite and infinite, and in this section we discuss these concepts in detail. 1. Finite sets and cardinality For a positive natural number n 1, set n = {1, 2, 3, . . , n}. If n = 0, let 0 = ∅. Then the set n has n elements and we can think of it as the standard set of that size. 1. Let f : X −→ Y be a function. • f is an injection or one-one (1-1 ) if for x1 , x2 ∈ X, f (x1 ) = f (x2 ) =⇒ x1 = x2 . • f is a surjection or onto if for each y ∈ Y , there is an x ∈ X such that y = f (x).

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