# MATH 4550 - Modern Algebra I

Class information:

Notes:

• Notes on the dihedral group can be found here.
• Lemma about the identity element and inverses in subgroups.
• Lemma about what cyclic subgroups look like.
• The division algorithm statement and proof
• Classification of homomorphisms from cyclic groups: lemma and main theorem.
• A proof that the set of permutations of a set is a group under composition.

Homework:

• Homework #1: problems and solutions
See this for a better way to do the solutions for 2,3,13
Typos: The solutions on problem 5 should say "Thus, e = 2 and e = 1.   This is a contradition."

• Homework #2: problems and solutions
See this for a better way to do #2,9,10 and for two extra problems to do

• Homework #3: problems and solutions.
Typo: At the end of the proof 5(a) it says "By the lemma, k divides the order of phi(x)."   It should say "By the lemma, the order of phi(x) divides k."

• Homework #4: problems and solutions.
Typos: In the solutions for problem 6 I found the homomorphisms from Z_6 to Z_8.   It should have been Z_8 to Z_6 as stated in the problem set.   But my solution is correct for the Z_6 to Z_8 problem.   Try doing Z_8 to Z_6 in the same way.

• Homework #5: problems and solutions.
• Homework #6: problems and solutions.
• Homework #7: problems and solutions.
• Homework #8: problems and solutions.
• Homework #9: problems and solutions.

Student notes from F16 (Thanks Cynthia!):