Homework 07 - MATH 74 HOMEWORK 7 “If you don’t like your analyst see your local algebraist!” –Gert Almkvist Due Wednesday April 8th at

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: MATH 74 HOMEWORK 7 “If you don’t like your analyst, see your local algebraist!” –Gert Almkvist Due Wednesday April 8th at 3:10pm. Throughout, let G be a group. Many of the following problems are from (or adapted from) Herestein’s Topics in Algebra . (1) In the following, decide and supply reasoning as to whether the given systems are groups. If they aren’t point out an axiom that fails. (a) the set of integers with a · b := a- b . (b) the set of positive integers with a · b := ab (the usual product of integers). (c) the set of all rational numbers with odd denominators, with the usual additional of rational numbers. (2) Given a,b ∈ G , prove that the equations a · x = b and y · a = b have unique solutions for x,y ∈ G . In particular, show we have cancellation laws: a · u = a · w = ⇒ u = w, u · a = w · a = ⇒ u = w. (3) For an element of a group, let a n denote the n-fold multiplication of a with itself. If G is a group with ( a · b ) 2 = a 2 · b 2 , prove that...
View Full Document

This note was uploaded on 05/04/2011 for the course MATH 73 taught by Professor Danberwick during the Spring '09 term at University of California, Berkeley.

Ask a homework question - tutors are online