INSTRUCTOR’S
SOLUTIONS MANUAL
JOHN B. FRALEIGH AND NEAL BRAND
A FIRST COURSE IN
ABSTRACT ALGEBRA
EIGHTH EDITION
John B. Fraleigh
University of Rhode Island
Neal Brand
University of North Texas
,The author and publisher of this book have used their best efforts in preparing this book.
These efforts include the development, research, and testing of the theories and programs to
determine their effectiveness. The author and publisher make no warranty of any kind,
expressed or implied, with regard to these programs or the documentation contained in this
book. The author and publisher shall not be liable in any event for incidental or
consequential damages in connection with, or arising out of, the furnishing, performance,
or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2021, 2003 by Pearson Education, Inc. 221 River Street, Hoboken, NJ 07030.
All rights reserved.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission of the publisher. Printed in the
United States of America.
ISBN-13: 978-0-32-139037-0
ISBN-10: 0-321-39037-7
,Preface for Seventh Edition
This manual contains solutions to all exercises in the text, except those odd-numbered
exercises for which fairly lengthy complete solutions are given in the answers at the back
of the text. Then reference is simply given to the text answers to save typing.
I prepared these solutions myself. While I tried to be accurate, there are sure to be the
inevitable mistakes and typos. An author reading proof tends to see what he or she wants to
see. However, the instructor should find this manual adequate for the purpose for which
it is intended.
Morgan, Vermont J.B.F
July, 2002
Preface for Eighth Edition
In keeping with the seventh edition, this manual contains solutions to all exercises in the
text except for some of the odd-numbered exercises whose solutions are in the back of the
text book. I made few changes to solutions to exercises that were in the seventh edition.
However, solutions to new exercises do not always include as much detail as would be found
in the seventh edition. My thinking is that instructors teaching the class would use the
solution manual to see the idea behind a solution and they would easily fill in the routine
details.
As in the seventh edition, I tried to be accurate. However, there are sure to be some
errors. I hope instructors find the manual helpful.
Denton, Texas N.B.
March, 2020
, CONTENTS
0. Sets and Relations 01
I. Groups and Subgroups
1. Binary Operations 05
2. Groups 08
3. Abelian Examples 14
4. Nonabelian Examples 19
5. Subgroups 22
6. Cyclic Groups 27
7. Generators and Cayley Digraphs 32
II. Structure of Groups
8. Groups of Permutations 34
9. Finitely Generated Abelian Groups 40
10. Cosets and the Theorem of Lagrange 45
11. Plane Isometries 50
III. Homomorphisms and Factor Groups
12. Factor Groups 53
13. Factor Group Computations and Simple Groups 58
14. Group Action on a Set 65
15. Applications of G-Sets to Counting 70
VI. Advanced Group Theory
16. Isomorphism Theorems 73
17. Sylow Theorems 75
18. Series of Groups 80
19. Free Abelian Groups 85
20. Free Groups 88
21. Group Presentations 91
SOLUTIONS MANUAL
JOHN B. FRALEIGH AND NEAL BRAND
A FIRST COURSE IN
ABSTRACT ALGEBRA
EIGHTH EDITION
John B. Fraleigh
University of Rhode Island
Neal Brand
University of North Texas
,The author and publisher of this book have used their best efforts in preparing this book.
These efforts include the development, research, and testing of the theories and programs to
determine their effectiveness. The author and publisher make no warranty of any kind,
expressed or implied, with regard to these programs or the documentation contained in this
book. The author and publisher shall not be liable in any event for incidental or
consequential damages in connection with, or arising out of, the furnishing, performance,
or use of these programs.
Reproduced by Pearson from electronic files supplied by the author.
Copyright © 2021, 2003 by Pearson Education, Inc. 221 River Street, Hoboken, NJ 07030.
All rights reserved.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording, or otherwise, without the prior written permission of the publisher. Printed in the
United States of America.
ISBN-13: 978-0-32-139037-0
ISBN-10: 0-321-39037-7
,Preface for Seventh Edition
This manual contains solutions to all exercises in the text, except those odd-numbered
exercises for which fairly lengthy complete solutions are given in the answers at the back
of the text. Then reference is simply given to the text answers to save typing.
I prepared these solutions myself. While I tried to be accurate, there are sure to be the
inevitable mistakes and typos. An author reading proof tends to see what he or she wants to
see. However, the instructor should find this manual adequate for the purpose for which
it is intended.
Morgan, Vermont J.B.F
July, 2002
Preface for Eighth Edition
In keeping with the seventh edition, this manual contains solutions to all exercises in the
text except for some of the odd-numbered exercises whose solutions are in the back of the
text book. I made few changes to solutions to exercises that were in the seventh edition.
However, solutions to new exercises do not always include as much detail as would be found
in the seventh edition. My thinking is that instructors teaching the class would use the
solution manual to see the idea behind a solution and they would easily fill in the routine
details.
As in the seventh edition, I tried to be accurate. However, there are sure to be some
errors. I hope instructors find the manual helpful.
Denton, Texas N.B.
March, 2020
, CONTENTS
0. Sets and Relations 01
I. Groups and Subgroups
1. Binary Operations 05
2. Groups 08
3. Abelian Examples 14
4. Nonabelian Examples 19
5. Subgroups 22
6. Cyclic Groups 27
7. Generators and Cayley Digraphs 32
II. Structure of Groups
8. Groups of Permutations 34
9. Finitely Generated Abelian Groups 40
10. Cosets and the Theorem of Lagrange 45
11. Plane Isometries 50
III. Homomorphisms and Factor Groups
12. Factor Groups 53
13. Factor Group Computations and Simple Groups 58
14. Group Action on a Set 65
15. Applications of G-Sets to Counting 70
VI. Advanced Group Theory
16. Isomorphism Theorems 73
17. Sylow Theorems 75
18. Series of Groups 80
19. Free Abelian Groups 85
20. Free Groups 88
21. Group Presentations 91
