MAT 423 Assignments

Highlighted assignments are due soon.

Due Date

 Assignments
19 Nov
  • p. 123 #21 (parts ii-v), #22 (find only the inverse), #25 (already done, really — find in notes & adapt), #34 (look for non-disjoint transpositions)
  • Compute a Cayley table for \(S_3 = \{ (1), (1\,2), (1\,3), (2\,3), (1\,2\,3), (1\,3\,2) \} \).
  • Recall the set \(\Gamma_{20}=\{1,\zeta,\zeta^2,\ldots,\zeta^{19}\}\) where \[\zeta=\cos\left(\frac{2\pi}{20}\right)+i\sin\left(\frac{2\pi}{20}\right).\] This is a cyclic group under multiplication. Determine all its generators. The corollary near the end of class tells you how many there are; the theorem right before it tells you how to find them.
12 Nov Read Section 2.4. Be sure you know what a cyclic group is.
7 Nov p. 146 #36(i-v, viii, x), 37, 40, 41, 42, 44, 48
Note: I made at least one of those extra credit, probably 48 but I’m not sure.
31 Oct
(BOO!)		 Show that for any group \(G\) and any \(a\in G\), and for any \(m,n\in\mathbb Z\), \((a^m)^n=a^{mn}\) and \(a^ma^n=a^{m+n}\).
24 Oct Compute a Cayley table for \(D_3\). Don’t use geometry. That would take too long! Use only the facts we determined in class; namely,
  • \(D_3=\{\iota, \rho, \rho^2, \varphi, \rho\varphi, \rho^2\varphi \}\)
  • the operation is composition of functions
  • \(\iota\) is a “do-nothing” function where every point ends up in its original place
  • \(\rho^3=\varphi^2=\iota\)
  • \(\varphi\rho=\rho^2\varphi\), so we never need to write \(\varphi\) before \(\rho\)
15 Oct Exercises 1-5 on the handout on RSA encryption and decryption
10 Oct p. 74 #77, 80, 81, 84, 87, 88, 89, 91, 95, 96
8 Oct Read Chapter 1, Section 5 on RSA (pgs. 72-74) and the first few pages (5 or so) of Chapter 2, Section 3 on Groups
3 Oct p. 58 #68, 69, 70
26 Sept p. 53 #46, 47, 50, 56, 57, 60, 61
19 Sept p. 14 #1-4, 9, 11, 12, 15; read but do not do 22, 23, 26 (you can do them if you want, but no extra credit, sorry)
12 Sept Read Chapter 1, Section 3 (Greatest Common Divisors). Take notes on things that look important. There could be a quiz!
5 Sept Read Chapter 1, Section 1 (Induction). Be sure to take notes on things you see that I did not discuss in class.