MATH 430, MODERN ALGEBRA
SPRING 2011
1. A subgroup of a cyclic group is cyclic (the group is not necessarily nite).
2. If G is a group and H is a subgroup of G, then aH = bH if and only if a1 b H.
3. If G is a nite group, then the order of every subgroup
MATH 430, MODERN ALGEBRA
SPRING 2011
1. (a) Show that in the ring Zn , [a] has a multiplicative inverse if and only if a and
n are relatively prime (you can use the fact that if a and n are two non-zero integers,
then 1 can be written as a combination
1 =
MATH 430, MODERN ALGEBRA, SPRING 2011
SOLUTIONS TO THE SELECTED PROBLEMS, PROBLEM SETS 1-3
1. Exercises from the book:
4. 32: If a G, then since a a = e, a = a1 . Now for a, b G, we
have (a b) (a b) = e, so a (b a) b = e, so (b a) b = a1 , so
b a = a1 b1
MATH 430, MODERN ALGEBRA, SPRING 2011
SOLUTIONS TO THE SELECTED PROBLEMS, PROBLEM SETS 4,5
1. Exercises from the book:
14. 39: We dene so that it sends the left coset of H in G generated
by a to the left coset of H in G generated by (a) (the only natural
MATH 430, MODERN ALGEBRA, SPRING 2011
SOLUTIONS TO THE SELECTED PROBLEMS, PROBLEM SETS 6, 7
1. Exercises from the book:
18.40: (a) Assume to contrary that there is an isomorphism : 2Z
3Z, and let b = (2). Then
(4) = (2) + (2) = 2b.
And
(4) = (2)(2) = b2