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Austin Christian

Georgia Tech Mathematics

GT Summer Edge Program: Algebra (2024)

Instructor: Austin Christian

office hours: MTR 11am-12pm, Skiles 022

TA: Junzhe Mao

office hours: M 3pm-4pm, Skiles 146A

Meeting time: MTWR 9:00am-9:50am

Meeting location: Skiles 114

a gif showing the transformations r, ar, rar, and arar applied to a triangle

Announcements

For at least the first few weeks of the summer, our primary text will be Judson's Abstract Algebra: Theory and Applications. You are encouraged to familiarize yourself with chapters 1 and 2 before we get started, and should complete the exercises listed below before our first day.

Schedule

The following is a tentative, nonbinding schedule. It will be updated as the summer progresses. You should attempt all listed exercises, and submit solutions to those with stars.

	Topics covered	Notes	Exercises
Week 1 (5/27-5/30)	M: Memorial Day T: Intros and definition of group W: Examples and basic properties of groups R: Subgroups, subgroup criteria, cyclic subgroups	May 29 May 30	Due 5/28: The following exercises from Judson. 1.4: 25, 29 2.4: 1, 5, 16, 17, 20 Due 5/30: The following exercises from Judson. 3.5: 27, 28, 31, 48, 53 4.5: 1, 6, 10, 13, 20, 31*, 46
Week 2 (6/3-6/6)	M: Cyclic subgroups, permutation groups T: Subgroups of \(S_n\) W: Cosets R: Theorems of Lagrange, Euler, and Fermat	June 3 June 4 June 5 June 6	Due 6/4: The following exercises from Judson. 4.5: 6, 13, 20, 39 5.4: 1, 3, 4, 5 Due 6/6: The following exercises from Judson. 5.4: 12, 14, 23, 31, 32, 33, 34, 35* 6.5: 3, 4, 6*
Week 3 (6/10-6/13)	M: Homomorphisms, isomorphisms, and Cayley's theorem T: External and internal direct products W: Normal subgroups R: The first isomorphism theorem	June 10 June 11 June 12 June 13	Due 6/11: The following exercises from Judson. 6.5: 9, 12, 15, 16, 21, 23 9.4: 1, 2, 3, 8-10, 14, 15, 16 Due 6/13: The following exercises from Judson. 9.4: 17, 18, 19, 37, 41, 42 10.4: 2, 3, 4, 8, 9, 10, 13, 14
Week 4 (6/17-6/20)	M: More isomorphism theorems T: Classification of finite abelian groups W: Juneteenth R: Composition series	June 17 June 18 June 20	Due 6/18: The following exercises from Judson. 11.4: 2, 5, 6, 7, 9, 10, 14, 16, 19* 11.5: 2, 3, 4, 5, 6, 7, 9-11, 12 Due 6/20: The following exercises from Judson. 13.4: 3, 6, 7, 8, 11, 17*
Week 5 (6/24-6/27)	M: The Jordan-Hölder theorem T: Group actions W: Burnside's counting theorem R: The Sylow theorems F: Midterm exam (take home)	June 24 June 25 June 26 June 27	Due 6/25: The following exercises from Judson. 13.4: 9, 19, 20* 14.5: 1-4, 5* Due 6/27: The following exercises from Judson. 14.5: 7, 8, 9, 20, 21, 23
Week 6 (7/1-7/4)	M: Rings, integral domains, fields T: Subrings and ideals W: Maximal and prime ideals R: Fourth of July	July 1 July 2 July 3	Due 7/2: The following exercises from Judson. 15.4: 5, 6, 7, 8, 13, 16, 17, 19, 21, 22 Due 7/5: The following exercises from Judson. 16.7: 2, 5, 8, 12, 16, 20, 26, 27, 28, 33, 37
Week 7 (7/8-7/11)	M: Polynomial rings, day 1 T: Polynomial rings, day 2 W: Fields, day 1 R: Fields, day 2	July 8 July 9 July 10 July 11	Due 7/9: The following exercises from Judson. 16.7: 4, 5, 7, 8, 9, 39 17.5: 3, 4, 5, 7, 10* Due 7/11: The following exercises from Judson. 17.5: 11, 14, 18, 20, 21, 25, 26, 28 21.5: 1, 3, 4, 5*, 11, 14
Week 8 (7/15-7/18)	M: Galois theory, day 1 T: Galois theory, day 2 W: Galois theory, day 3 R: Final exam	July 15 July 16 July 17	Due 7/17: The following exercises from Judson. 21.5: 9, 11, 15, 22, 23 23.5: 1, 2, 5, 6, 7, 11, 14, 20