Homework2Solutions

Homework2Solutions - Homework 2 Solutions Math 332 Spring...

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: Homework 2 Solutions Math 332, Spring 2010 Problem 1. Proposition. Every finite group with an even number of elements has at least one element of order two. Proof. Let G be a finite group, and suppose that G does not have any elements of order two. We shall prove that the order of G is odd. Since G does not have any elements of order two, we know that a 6 = a- 1 for every non- identity element a ∈ G . If we pair each element of G with its inverse, we get a partition of G of the following form: { e } , a 1 ,a- 1 1 , a 2 ,a- 1 2 , ... a n ,a- 1 n . From this partition, we see that G must have 2 n + 1 elements, where n is the number of inverse pairs. In particular, the order of G must be odd. Problem 2. Proposition. Let G be a group, and let a be a fixed element of G . Define a new binary operation * on G by x * y = xay for all x,y ∈ G . Then G forms a group under the operation * . Proof. We must prove that * is associative and possesses an identity element, and that each element of G...
View Full Document

This document was uploaded on 11/03/2010.

Page1 / 3

Homework2Solutions - Homework 2 Solutions Math 332 Spring...

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