1. Compute the positive integer n as follows.

• Add up the digits in your student identity number.

• Call this number n unless it is prime, in which case add 1 to it and call the answer n.

(a) We consider all possible groups of order n.

(i) How many such groups are there? [Quote your source.]

(ii) How many of those groups are abelian? [Quote your source.]

(iii) For a group G of order n, what numbers might possibly occur as the orders of the elements of G? Can we be sure that any of those numbers will definitely occur as the order of an element of the group? [Justify your answer.]

(iv) For a group G of order n, what numbers might possibly occur as the orders of the subgroups of G? Can we be sure that any of those numbers will definitely occur as the order of a subgroup of the group? [Justify your answer.]

(b) Consider the symmetric group Sn of all permutations of the set X = {1, 2, 3, . . . , n}.

(i) What is the order of Sn?

(ii) How many of the permutations represent symmetries of a regular n-gon?

(iii) Describe all the symmetries of a regular n-gon. For those that are rotations, specify the angle of rotation in each case. For those that are reflections, specify the axis about which each reflection takes place.