Problem 1. Let G be a group. Show that for every element a G, (a 1 ) 1 = a.

Problem 2. Let (G, ) be a group. Show that if a c = b c, then a = b.

Problem 3. Let G denote a group with 4 members: denote these members by e (the identity element of G), and a, b, c. Suppose that a a = b and a b = c. Then give the operation table for this group. (Here denotes the operation of the group and x y denotes the product of two elements x and y. Is this group abelian?

Problem 4. Consider the group S(3). Show that there are four elements satisfying 2 = id and three elements satisfying 3 = id.

Problem 6. Let G be a group. (1) Define a relation on G by setting a b if and only if a = b 1 . Use Problem 1 to show that this relation is symmetric. (2) Show that if G has an even number of elements, then there is an element a 6= e such that a = a 1 . [Hint: consider the partition of G associated to the relation defined in the first part of this problem (you might want to go back and look at Thm 2.3.1). How many elements can there be in each equivalence class?] (3) Deduce from the point above that if G has an even number of elements, then there is an a G, with a 6= e such that a 2 = e.