Algebra Assignment 1

1    1.1    18, 22

2    1.1    20, 23

3    1.1    26, 27

4    1.3    12, 15, 19

5    Let  H  and  K  be subgroups of a group G.  Show that the union of  H  and  K  is
      a subgoup of  G  if and only if either  H  is contained in  K  or  K  is contained in H.

6    Let  G  be a group.
        a    Suppose  x² = 1  for all  x  in  G.  Show that  G  is abelian.
        b    Let  H  be a proper subgroup of  G.  Suppose that  x² = 1  for all  x  that are
              in  G  but not in  H.  Show that  H  is abelian.