Math 5331 Abstract Algebra I

Dr. Cordero

__Problems in
group theory I__

- Prove that if G is a group, , and for some , then must be the identity element of G.
- Prove that a group is abelian if each of its nonidentity elements has order 2.
- Prove or give a counterexample: If a group G has a subgroup of order n, then G has an element of order n.
- Prove that if A and B are subgroups of a group G, and is also a subgroup, then or .
- 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. - Prove that if G is a group of order (p a prime) and G is not cyclic, then for each .
- Prove that if H is a subgroup of a group G, [G:H]=2, , , then .
- 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 .
- Show that if G is an abelian group, then defined by for each is an automorphism of G.