practice2c - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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

View Full Document Right Arrow Icon
MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS February 24, 2007 Here are a few practice problems on groups. Try to work these WITHOUT looking at the solutions! After you write your own solution, you can compare to my solution. Your solution does not need to be identical—there are often many ways to solve a problem—but it does need to be CORRECT. 1. Suppose that N is a normal subgroup of a group G . Show that if a G has ±nite order o ( a ), then Na in G/N has ±nite order m with m | o ( a ). Solution Let n = o ( a ). Then a n = e , so ( Na ) n = Na n = Ne = N , which is the identity element of G/N . Hence we must have o ( Na ) | n . ± Question: Will it be true that o ( Na ) = o ( a )? (No.) Try an example. 2. a. Suppose that H and K are groups. I won’t prove it here, but you should be able to prove that the Cartesian product H × K = { ( h, k ) : h H, k K } is a group with group operation ( h 1 , k 1 )( h 2 , k 2 ) = ( h 1 h 2 , k 1 k 2 ). Prove the following facts about this group. a. Show that
Background image of page 1

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

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

{[ snackBarMessage ]}

Page1 / 2

practice2c - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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