Abstract Algebra Exam 1 Review Problems and Solutions

#abstractalgebra #abstractalgebraexam #grouptheory

Links and resources
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(0:00) Introduction
(0:47) a divides b definition
(2:10) Euclid's Lemma
(4:01) Relatively prime definition
(5:33) Group definition
(10:22) Center of a group definition
(12:56) Isomorphism definition
(16:56) Are cyclic groups Abelian?
(18:46) Are Abelian groups cyclic?
(21:53) Is D3 (dihedral group) cyclic? (D3 is the symmetries of an equilateral triangle)
(26:47) GCD is a linear combination theorem
(27:30) If |a| = 6, is a^(-8) = a^(4)? (the order of "a" is 6)
(29:21) Do the permutations (1 3) and (2 4) commute? (they are disjoint cycles)
(30:21) Is the cycle (1 2 3 4) an even permutation?
(31:56) Number of elements of order 2 in S4, the symmetric group on 4 objects
(36:19) Generators of the cyclic group Z24. Relationship to U(24). Euler phi function value φ(24).
(42:06) If |a| = 60, answer questions about (a) (cyclic subgroup generated by a): possible orders of subgroups, elements of (a^12), order |a^12|, order |a^45|.
(49:17) Permutation calculations, including the order of the product of disjoint cycles as the lcm of their orders (least common multiple of their orders)
(54:51) One-step subgroup test to prove the stabilizer of an element under a permutation group is a subgroup of that permutation group.
(1:02:55) Induction proof that φ(a^n) = (φ(a))^n for all positive integers n.
(1:08:25) Direct image of a subgroup is a subgroup (one-step subgroup test).
(1:13:52) Prove a relation is an equivalence relation. Find equivalence classes. (Related to modular arithmetic).

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[TwistedFour]

Hello Dr. Kinney,

I'm studying for my exam right now and your video helped me out a lot.
I appreciate you taking your time to make this review guide for students.

Thank you and have a wonderful day.

[martinhawrylkiewicz]

Great vid! I took Abstract algebra as an undergrad a decade ago and really liked it but over the years I forgot a lot of things, so now I am starting to teach myself Abstract Algebra using a textbook by John Fraleigh (7th edition). I find it very refreshing and wonderful to be able to see abstract algebra again after all these years.

[brandonfox]

For the binary operation, you needed a Cartesian product for the first part. Every unique ordered pair (a, b) in G X G gets assigned to an element a * b in G.

[JamesWeeks-jm]

Hi Dr. Kinney,

Thanks for the excellent review. I am back in college after being gone for 4 years and the first proof-based math course I am taking (outside of Intro to Proofs) is Abstract Algebra. Preparing for exam in 2 weeks so this was amazing in making sure I understood some of the fundamental concepts, problems, proofs. It has been tough learning this subject and relearning complex numbers, basic linear algebra, but so far has been a great experience in learning the basics of Abstract Algebra.

[eddiejennings]

Thank you! A great gift to math students.

[smpelness]

Ive started college about three days ago and abstract algebra aswell, and this really helped me, thanks for what you do sir

[filipjohansen]

The quality of your content is phenomenal. It's quite the gift to the world

[ronmiller]

You are an excellent professor in Mathematics Professor Kinney. Keep up the excellent lectures!

[willgreen]

Thank you so much. I have always enjoyed learning mathematics. You teach very well and I am very thankful for your videos! Please keep them up!!

[richard-]

Thank you very much for sharing these lectures and examples. I’m a retired IT consultant who has always enjoyed maths and did some research a long time ago. Your lectures are very clear and these examples are useful and illuminating too.

[FerreroRocher-rf]

Thank you so much for this Sir, this is so helpful ❤

[ronnies.]

Wish I had this video when I was taking abstract! Thanks, Professor.

[mathematik]

I love this Review Exam....help so so so much!!!!

[oldPrince]

Well explained everything, good video!

[Simpuls]

Writing a comment on the things I learned in this video which I didn't know before:

1. cyclic is not equal to a group being generated by elements. For Question 9 I wasn't sure. Since D3 is in fact generated by just rotations and symmetries I thought, maybe that's the same as saying, it's cyclic, but it's not. Cyclic seems to mean, that one and only one element generates the Group.

[leonardomurgia]

10:01 Shouldn't it be  ∃! Instead of  ∃?

[joshbullis]

Hello Professor! I am about to take Abstract Algebra (undergrad). I have had Real Analysis and Topology already, but im still nervous about the new semester. Im also doing DiffEQ and Complex Analysis. Do you have any advice about how to manage those 3 courses in the same semester. Thanks!