Problems in group theory I

Math 5331 – Abstract Algebra I
Dr. Cordero

  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.