Math 417 – Seventeenth Day – Class, Study notes of Number Theory

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Math 417 – Seventeenth Day – Class

Bruce Reznick University of Illinois at Urbana-Champaign

October 2, 2020

Before I repeat the frames with the overview, I’d like to answer a few questions I’ve gotten overnight.

1. An extremely alert student noted that on frames from Sept. 9 (Day 7), I multiplied permutations in the wrong order. This was corrected on Sept. 11 (Day 8), so keep that in mind when studying.
2. A student writes to ask whether ”generators” will be on the test. What I said was:

Yes, they might be on the test. I might give you a cyclic group and one generator and ask you to find another generator, but it won’t involve large numbers: for example, I wouldn’t ask you to decide if [2] 37 is a generator of ((Z/ 37 Z)∗, ).

g. As always, read the problems carefully.

h. The exam is open book, open notes and open to the class notes I’ve been providing. You may use a calculator, but it shouldn’t be necessary.

i. This is a non-collaborative exam. Please do not talk to anyone else about the test unless you know that everyone in the conversation has taken it.

j. Despite the relaxed nature of these rules, I am attaching a link to §1-402 of the Student Code regarding Academic Integrity. I’ve read it, and I hope you have too.

https://studentcode.illinois.edu/article1/part4/1-402/

Here are the frames again. Please let me know if you have questions or if I missed something.

Number theory topics: Z, Q, divisibility, m | n, congruence mod n, a ≡ b mod n, [a]n, prime numbers, gcd, gcd(m, n), relatively prime integers, the existence of prime factorization, Euclidean Algorithm. Know what the Euler phi function φ(n) means, but you won’t have to calculate it. Know that gcd(m, n) = g implies that there exist integers r , s so that g = mr + ns (findable through the EA) and if gcd(m, n) = 1 and n | mr , then n | r.

Group Theory Vocabulary: commutative, associative, binary operations, the definition of a group, and an identity and inverses in a group, abelian groups, cyclic groups (Cn = 〈a〉, an^ = e or (Z/nZ, ⊕)), the symmetric group Sn in general, and in more detail S 3 , the Klein group V and the dihedral group D 4.

With permutations, know cycles and transpositions. Write permutations in multiple ways, for example if π : { 1 , 2 , 3 , 4 } → { 1 , 2 , 3 , 4 } and π(1) = 2, π(2) = 4, π(3) = 3, π(4) = 1, then we might write this as:

π : 1 7 → 2 7 → 4 7 → 1 , 3 7 → 3 ,

π = (124)(3), π = (124), π =

Know how to multiply permutations in the right order.

Not on this test φ(n) (except incidentally as |((Z/nZ)∗, )|), repeating decimals as such, Cayley’s Theorem, odd/even permutations, the book’s theorem on the classification of finite abelian groups. What we’ve done this week on Aut(G ), ig , etc.

Send me an email if there’s something I missed.