Question

1. Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category.
2. Consider the `sets of numbers' listed in §1, and decide which are made into groups by conventional operations such as + and . Even if the answer is negative (for example, (K, ) is not a group), see if variations on the definition of these sets lead to groups (for example, (R*, ) is a group cf. question 4.
3. Prove that (gh)^-1 = h^-1g^-1 for all elements g, h of a group G.
4. Suppose that g^2 = e for all elements g of a group G; prove that G is commutative.
5. In the group of invertible 2 x 2 matrices, consider g= (0 -1

        1      0) , h= (0     1

-1   -1)

       Verify that IgI = 4, IhI = 3, and Ighl = ∞

Answer 1