HW2_soln - Homework #2 Solutions 14. Suppose that G is a...

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

View Full Document Right Arrow Icon
Homework #2 Solutions 14. Suppose that G is a group with the following property: for any a, b, c G , ab = ca implies b = c . Let x, y G . Set a = x - 1 , b = xy and c = yx . Then ab = x - 1 ( xy ) = ( x - 1 x ) y = ey = y = ye = y ( xx - 1 ) = ( yx ) x - 1 = ca. By our hypothesis, we must have xy = b = c = yx . Since x and y were arbitrary, we conclude that G is abelian. 16. Let G be a group and let a, b G . Using the associativity property of groups we have ( ab )( b - 1 a - 1 ) = a ( bb - 1 ) a - 1 = aea - 1 = aa - 1 = e and ( b - 1 a - 1 )( ab ) = b ( aa - 1 ) b - 1 = beb - 1 = bb - 1 = e. Since inverses are unique, we must have ( ab ) - 1 = b - 1 a - 1 . Note: In class I showed that any one-sided inverse in a group is automatically a two-sided inverse. Therefore, any one of the above inequalities also establishes the result. 20. We will prove by induction that if G is a group, n Z + and a 1 , a 2 , . . . , a n G then ( a 1 a 2 ··· a n ) - 1 = a - 1 n a - 1 n - 1 ··· a - 1 2 a - 1 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/29/2012 for the course MATH 3362 taught by Professor Ryandaileda during the Fall '10 term at Trinity University.

Page1 / 2

HW2_soln - Homework #2 Solutions 14. Suppose that G is a...

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