Math 5331 Abstract Algebra I

Dr. Cordero

Problems in group theory I

1. Prove that if G is a group, , and  for some , then  must be the identity element of G.
2. Prove that a group is abelian if each of its nonidentity elements has order 2.
3. Prove or give a counterexample: If a group G has a subgroup of order n, then G has an element of order n.
4. Prove that if A and B are subgroups of a group G, and  is also a subgroup, then  or .
5. Assume that H and K are subgroups of a group G and that . The subset of G defined by is called a double coset of H and K in G. Prove that if  and  are double cosets of H and K in G, then they are either equal or disjoint.
6. Prove that if G is a group of order  (p a prime) and G is not cyclic, then  for each .
7. Prove that if H is a subgroup of a group G, [G:H]=2, , , then .
8. Prove that if A and B are finite subgroups of a group G, and |A| and |B| have no common divisor greater than 1, then .
9. Show that if G is an abelian group, then  defined by  for each  is an automorphism of G.