Abstract Algebra

Assignment #2

Kenneth Levasseur
Mathematical Sciences
UMass Lowell
Kenneth_Levasseur@uml.edu

Due  October 3.

Instructions: Write out your proofs using clear, concise logic.   Use separate sheets of paper for each problem.

1.  If G is an abelian group that contains a pair of cyclic subgroup s of order 2, show that G must contain a subgroup of order 4.   Does this subgroup have to be cyclic?   (This is #33 of Chapter 4 in the book)

2.   Prove that if G is a finite group, and H is a nonempty subset of G that is closed with respect to the operation of G (i. e., a,b ∈H=> a b∈H), then H is a subgroup of G.  (Compare with Proposition 3.9)

3.  What are the possible cycle structures of elements in (Subscript[A, 6])?  (This is the second half of #11 in Chapter 5 of the book.  The solution to the first half is in the solutions at the end of the book.)

Additional Notes

There are three video links, including two related to permutations at http://applied-discrete-structures.wiki.uml.edu/Chapter+15

Import["http://homepage.mac.com/klevasseur/Master.m"]

SubgroupGenerated[Dihedral[3],Ref]