MATH 417 Study Guide - Final Guide: Fermat Number, Division Ring, Integral Domain

• School: University of Illinois
• Department: Mathematics
• Course: MATH 417
• Professor: All

Document Summary

Math 417 spring 2017 section b1. Solutions: (50 points) consider the symmetric group in 4 elements s4 and let h s4 be the subgroup: Solution: (a) for any s4 we have. Hence, if h then 1 h, and we conclude that h is normal in a4. (b) by lagrange"s theorem: On the other hand, |a4/h| = 4!/8 = 3, and any group of order 3 is isomorphic to z3, hence is abelian. Solution: (a) a = r (real numbers); (b) a = h (quaternions); (c) a = z (integers); (d) a = 2z (even integers); (e) a = m2(r) (real 2 2 matrices). 3: (50 points) let (a, +, ) be an ordered ring with identity 1. De ne the absolute value of an element a a as usual by: Show that for a, b a the following statements are equivalent: (a) |a| |b|; (b) |b| a |b|; (c) a2 b2. |a| |b| = a |b| and a |b|