4107_final_2005 - Final Exam, Math 4107 December 12, 2005...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Final Exam, Math 4107 December 12, 2005 1. (Easy) Define the following terms: a. Group b. Ring c. Field d. Principal Ideal Domain e. Euclidean Domain 2. (Medium) In this problem you will construct a non-commutative group of order 21: We will take the set G defining our group to be Z 3 × Z 7 , and we will define a funny addition. This addition works by ( a, b ) + ( c, d ) = ( a + c (mod 3) , b · 2 c + d (mod 7)) . a. Prove that this is indeed a group. (Hint: Be careful in showing that the group operation is well-defined; also, associativity is slightly tricky. ) b. Prove that this group is non-commutative. c. (Optional +1 point extra credit) Why doesn’t this example work to show that there is a group of order 15 which is non-abelian? (We know all groups of order 15 are abelian). I could explain in about half a class where this example comes from. It is an example of a semi-direct product of Z 3 and Z 7 . Alas, the semester is over and our time has come to an end......
View Full Document

This note was uploaded on 10/23/2011 for the course MATH 4107 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.

Page1 / 3

4107_final_2005 - Final Exam, Math 4107 December 12, 2005...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online