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Contemporary Abstract Algebra
Joseph Gallian is a well-known active researcher and award-winning teacher. His Contemporary Abstract Algebra, 6/e, includes challenging topics in abstract algebra as well as numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings that give the subject a current feel and makes the content interesting and relevant for students.
640 pages, Hardcover
First published July 9, 2012
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Displaying 1 - 30 of 35 reviews
September 7, 2017
Before I delve into an analysis of this excellent textbook, let me highlight first that abstract algebra, contrary to much uniformed opinion, is not a specialistic, esoteric field in pure mathematics characterized by very limited applicability to the physical world. On the contrary, this subject is not just extremely beautiful, but a very important and even foundational discipline in many areas of mathematics and science.
In fact, structure and symmetry are a fundamental aspect of physical reality. It might well be argued that science is nothing but the discovery of symmetries, patterns and structures that define reality at its most fundamental level.
There is for example a natural connection between particle physics and representation theory of abstract algebra, linking the properties of elementary particles to the structure of Lie Groups and Lie Algebras (in particular, the different quantum states of any given elementary particle corresponds to an irreducible representation of the Poincare' group). Poincare' invariance is the fundamental symmetry in particle physics: a relativistic quantum field theory must have a Poincare'-invariant action (one of the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is represented by the Poincare Group, the symmetry group of special relativity).
We should also not forget that, while the spacetime symmetries represented by the the Poincare' group are the first that come to mind and the easiest to conceptualize, the physical world is also defined by many other types of symmetries, not least some internal symmetries such as the “color� symmetry SU(3) (the symmetry corresponding to the continuous interchange of the three quark “colors� responsible for the strong force). Moreover, a gauge U(2) × U(1) theory fully explains the unified electro-weak interactions (the requirement that a theory be invariant under local gauge transformations involving the phase of the wave function ultimately brought to the positing of the laws underpinning the interaction of light and matter - indeed, in quantum mechanics, gauge symmetry can be seen as the basis for electromagnetism and conservation of charge).
In more general term, local symmetries play a fundamental role in physics as they form the basis for gauge theories (a gauge theory is essentially a type of field theory "in which the Lagrangian is invariant under some Lie groups of local transformations").
It is also important to highlight that symmetry in general is fundamental in the physical world: as an example charge, parity, and time reversal symmetries (CPT) are fundamental symmetries in the Standard Model, whose transformations are symmetric under the simultaneous operation of charge conjugation (C), parity transformation (P), and time reversal (T).
Going beyond the realm of fundamental physics, molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties such as its dipole moment. In the biological sciences, I was told (my personal knowledge of the biological sciences is pretty rudimentary) that symmetry and symmetry breaking play a prominent role in developmental biology, from bilaterians to radially symmetric organisms (see for example ).
Moreover, group theory has many applications in cryptography, robotics and computer vision, just as another example of concrete application of abstract algebra concepts.
Abstract algebra is therefore extremely important and foundational in many fields, and not just in pure mathematics: it provides a set of powerful technical and conceptual tools that allow abstraction of apparently complex systems in such a way as to allow a rigorous mathematical treatment of some of their fundamental and defining features (symmetries and patterns) through the usage of algebraic structures such as groups, rings, vector spaces, algebras and fields. Adding hours on a clock, for example, is like working in a cyclic group; many manufacturing processes might be shown to be isomorphic to products of permutations of a finite group; and group theory and abstract algebra applied to molecular systems biology support the design of new drugs. In general, abstract algebra (and group theory in particular) provides a framework for constructing analogies or models from abstractions, and for the manipulation of those abstractions to design new systems, make predictions and posit new hypotheses.
OK, now that justice has been rendered to this fascinating and beautiful subject, the time has finally come for me to review this informative University textbook, which represents a very good introductory treatment of the fundamental elements of abstract algebra (group theory, rings, fields) at a level which is typically addressed at senior Mathematics undergraduate stage.
This book is very accurate and nicely written, with very few typos (all of them minor), a very good choice and variety of relevant exercises at different level of complexity (the majority of them are actually pretty easy), with minimal and only occasional hand-waving; it is also a book rich with examples (a critical feature in a subject that sometimes can sorely test the abstract thinking capabilities of even the keenest reader), and presenting a tightly organized and generally reader-friendly progression of concepts and techniques. I also greatly appreciated the full list of notational conventions at the beginning of the book, and the list of suggested readings at the end of each chapter.
On the not-so-positive side, I must said that I would have preferred way more depth and detail in some areas such as symmetry groups, that the proofs of some theorems are a bit too terse (leaving conceptual and computational gaps that require a bit of effort to fill), and that too few examples of practical applications are presented, occasionally giving the book a very “abstract� feeling. I must also point out, though, that this book contains one of the best treatments of factor groups, the Lagrange theorem, the Sylow theorems and the fundamental theorem of finite abelian groups that I have come across so far.
In summary, my general experience with this book has been quite a positive one; independently of the quality of this particular book, I must also say that studying abstract algebra has been for me a rewarding intellectual journey (I did study some abstract algebra at university, but it was only at introductory level and manly focused on vector spaces), as it is very remarkable how, by starting just with the basic definition of an extremely simple concept such as that of a group, whole worlds of progressively richer and more complex structures, patterns and relationships are progressively unveiled in a process of enthralling discovery.
Overall, this is a 4-star book. Not a perfect book by any means, and possibly too basic in some areas, but a solid good textbook recommended to anybody interested in a treatment of this fascinating subject delivered, with a good pedagogical approach, at senior undergraduate level.
In fact, structure and symmetry are a fundamental aspect of physical reality. It might well be argued that science is nothing but the discovery of symmetries, patterns and structures that define reality at its most fundamental level.
There is for example a natural connection between particle physics and representation theory of abstract algebra, linking the properties of elementary particles to the structure of Lie Groups and Lie Algebras (in particular, the different quantum states of any given elementary particle corresponds to an irreducible representation of the Poincare' group). Poincare' invariance is the fundamental symmetry in particle physics: a relativistic quantum field theory must have a Poincare'-invariant action (one of the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is represented by the Poincare Group, the symmetry group of special relativity).
We should also not forget that, while the spacetime symmetries represented by the the Poincare' group are the first that come to mind and the easiest to conceptualize, the physical world is also defined by many other types of symmetries, not least some internal symmetries such as the “color� symmetry SU(3) (the symmetry corresponding to the continuous interchange of the three quark “colors� responsible for the strong force). Moreover, a gauge U(2) × U(1) theory fully explains the unified electro-weak interactions (the requirement that a theory be invariant under local gauge transformations involving the phase of the wave function ultimately brought to the positing of the laws underpinning the interaction of light and matter - indeed, in quantum mechanics, gauge symmetry can be seen as the basis for electromagnetism and conservation of charge).
In more general term, local symmetries play a fundamental role in physics as they form the basis for gauge theories (a gauge theory is essentially a type of field theory "in which the Lagrangian is invariant under some Lie groups of local transformations").
It is also important to highlight that symmetry in general is fundamental in the physical world: as an example charge, parity, and time reversal symmetries (CPT) are fundamental symmetries in the Standard Model, whose transformations are symmetric under the simultaneous operation of charge conjugation (C), parity transformation (P), and time reversal (T).
Going beyond the realm of fundamental physics, molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties such as its dipole moment. In the biological sciences, I was told (my personal knowledge of the biological sciences is pretty rudimentary) that symmetry and symmetry breaking play a prominent role in developmental biology, from bilaterians to radially symmetric organisms (see for example ).
Moreover, group theory has many applications in cryptography, robotics and computer vision, just as another example of concrete application of abstract algebra concepts.
Abstract algebra is therefore extremely important and foundational in many fields, and not just in pure mathematics: it provides a set of powerful technical and conceptual tools that allow abstraction of apparently complex systems in such a way as to allow a rigorous mathematical treatment of some of their fundamental and defining features (symmetries and patterns) through the usage of algebraic structures such as groups, rings, vector spaces, algebras and fields. Adding hours on a clock, for example, is like working in a cyclic group; many manufacturing processes might be shown to be isomorphic to products of permutations of a finite group; and group theory and abstract algebra applied to molecular systems biology support the design of new drugs. In general, abstract algebra (and group theory in particular) provides a framework for constructing analogies or models from abstractions, and for the manipulation of those abstractions to design new systems, make predictions and posit new hypotheses.
OK, now that justice has been rendered to this fascinating and beautiful subject, the time has finally come for me to review this informative University textbook, which represents a very good introductory treatment of the fundamental elements of abstract algebra (group theory, rings, fields) at a level which is typically addressed at senior Mathematics undergraduate stage.
This book is very accurate and nicely written, with very few typos (all of them minor), a very good choice and variety of relevant exercises at different level of complexity (the majority of them are actually pretty easy), with minimal and only occasional hand-waving; it is also a book rich with examples (a critical feature in a subject that sometimes can sorely test the abstract thinking capabilities of even the keenest reader), and presenting a tightly organized and generally reader-friendly progression of concepts and techniques. I also greatly appreciated the full list of notational conventions at the beginning of the book, and the list of suggested readings at the end of each chapter.
On the not-so-positive side, I must said that I would have preferred way more depth and detail in some areas such as symmetry groups, that the proofs of some theorems are a bit too terse (leaving conceptual and computational gaps that require a bit of effort to fill), and that too few examples of practical applications are presented, occasionally giving the book a very “abstract� feeling. I must also point out, though, that this book contains one of the best treatments of factor groups, the Lagrange theorem, the Sylow theorems and the fundamental theorem of finite abelian groups that I have come across so far.
In summary, my general experience with this book has been quite a positive one; independently of the quality of this particular book, I must also say that studying abstract algebra has been for me a rewarding intellectual journey (I did study some abstract algebra at university, but it was only at introductory level and manly focused on vector spaces), as it is very remarkable how, by starting just with the basic definition of an extremely simple concept such as that of a group, whole worlds of progressively richer and more complex structures, patterns and relationships are progressively unveiled in a process of enthralling discovery.
Overall, this is a 4-star book. Not a perfect book by any means, and possibly too basic in some areas, but a solid good textbook recommended to anybody interested in a treatment of this fascinating subject delivered, with a good pedagogical approach, at senior undergraduate level.
August 21, 2009
This book is OK as far as presenting abstract algebra in the usual way to undergrads. Competent explanations of the basics of groups, rings, and fields. Numerous easy exercises, which is fine, although it might be nice if there were more challenging ones too.
There are two problems.
The first problem is, I don't believe in the purely abstract approach to teaching a first course in algebra which this book uses. This book doesn't give you any real idea what the heck algebra is good for and doesn't provide any real connections to anything else. The attempts to liven things up with silly quotes and bios give the book a condescending middle school type feel. I would rather the author attract readers' interest by showing them the power of the algebraic structures discussed. Groups and rings are extremely powerful concepts, but to read this book you would think they are just a game where you write down some axioms and see what random statements you can prove. (None of this is unique to this book. It seems students are simply not expected to see algebra put to any use until grad school, if at all.)
The second problem is, this book sells for $170. There is no excuse for this. It is simply disgusting. No one should buy this book or require it for a course at such an exorbitant price. It doesn't do your laundry or cook you breakfast. A $12 Dover book would do fine.
There are two problems.
The first problem is, I don't believe in the purely abstract approach to teaching a first course in algebra which this book uses. This book doesn't give you any real idea what the heck algebra is good for and doesn't provide any real connections to anything else. The attempts to liven things up with silly quotes and bios give the book a condescending middle school type feel. I would rather the author attract readers' interest by showing them the power of the algebraic structures discussed. Groups and rings are extremely powerful concepts, but to read this book you would think they are just a game where you write down some axioms and see what random statements you can prove. (None of this is unique to this book. It seems students are simply not expected to see algebra put to any use until grad school, if at all.)
The second problem is, this book sells for $170. There is no excuse for this. It is simply disgusting. No one should buy this book or require it for a course at such an exorbitant price. It doesn't do your laundry or cook you breakfast. A $12 Dover book would do fine.
March 20, 2010
My favorite textbook of all time!
May 25, 2010
This book is the reason I became a mathematician.
January 12, 2012
This book changed my life!
February 12, 2012
This book is the reason I became a mathematician.
June 30, 2021
This is not a book for beginners, yet those that teach algebra like this book and those that praise this book often cite its library of exercises.
Gallian suffers the expert's dilemma.
He is verbose where he could be concise (e.g. don't apply modulo to an exponent when working with polynomials and modding their coefficients) and terse where he should elaborate (e.g. the definition of SL, or the U_x(n), or how to draw a find polynomial zeros, or how to construct a basis for a field or...).
There are a ton of exercises, where too much of important material is imparted. This is a bridge to upper level math texts, where all the true learning is in the problems, but the presentation is of a calculation-heavy text like you have in grade school. It's thereby constantly at odds with itself.
Gallian regularly defines terms, conditions, and notation in prose - only sometimes italicizes - rather than emphasizing or calling out with a colored box important terms. Definitions aren't numbered, so you need to keep a hand-written glossary of terms for yourself. Even then, the wording of the definition is too often weirdly self referential. The index is pathetic.
Theorems are typically stated clearly, almost all important theorems are proven as a part of the main text, with only a handful left as exercises. Thankfully, many exercises for the reader are actually in the exercises section, which are usually ordered from concrete to abstract: this allows one to work up to a more abstarct problem from the concrete.
For example, exercise one might have be list the elements of a given group. Exercise 2 might have a slightly larger group, and exercise 3 larger still. Exercise 4 is then a proof of the size (order) of the group given an arbitrary (valid) argument or about its structure. The exercises in the Linear Algebra chapter (chapter 19 in the 9th edition) were solid.
Why 2/5 rather than 1/5? There are good things here. Groups were covered pretty well and I ingested enough material to actually prove a few things independent of the sketch-as-proof solution manual (that gave more direction than the weirdly worded exercise). The coverage of rings was long-winded and shallow. Fields were taught hand-in-hand with rings so it's unclear where ring theory stops and field theory starts.
The biggest problem with this text is that very little of this material feels like it matters outside of intellectual showboating. Thanks to the coverage of symmetry groups, I'm looking at my Rubik's cube collection in a whole new way - and I love it, honestly - but that's it.
I know some loose things abstractly, but the book didn't covey why. My love of history was met with brief biographies on important women and men of Abstract Algebra's history. I can understand Andrew Wiles proof of Fermat's Last Theorem and can read Erno Rubik's work on group theory without being a chump.
But I don't have the capacity or good will to recommend this text or this subject to anyone to doesn't care about those two things.
Oh, or if you need to see lots of curious ways isomorphism is implied between two groups, this will help you out.
There are better, clearer, FREE texts on the internet covering this subject.
Don't waste your money, even if it's a rental.
Note: I didn't read all of the special topics, but those might as well be Appendices given they read so different from the rest of the text
Gallian suffers the expert's dilemma.
He is verbose where he could be concise (e.g. don't apply modulo to an exponent when working with polynomials and modding their coefficients) and terse where he should elaborate (e.g. the definition of SL, or the U_x(n), or how to draw a find polynomial zeros, or how to construct a basis for a field or...).
There are a ton of exercises, where too much of important material is imparted. This is a bridge to upper level math texts, where all the true learning is in the problems, but the presentation is of a calculation-heavy text like you have in grade school. It's thereby constantly at odds with itself.
Gallian regularly defines terms, conditions, and notation in prose - only sometimes italicizes - rather than emphasizing or calling out with a colored box important terms. Definitions aren't numbered, so you need to keep a hand-written glossary of terms for yourself. Even then, the wording of the definition is too often weirdly self referential. The index is pathetic.
Theorems are typically stated clearly, almost all important theorems are proven as a part of the main text, with only a handful left as exercises. Thankfully, many exercises for the reader are actually in the exercises section, which are usually ordered from concrete to abstract: this allows one to work up to a more abstarct problem from the concrete.
For example, exercise one might have be list the elements of a given group. Exercise 2 might have a slightly larger group, and exercise 3 larger still. Exercise 4 is then a proof of the size (order) of the group given an arbitrary (valid) argument or about its structure. The exercises in the Linear Algebra chapter (chapter 19 in the 9th edition) were solid.
Why 2/5 rather than 1/5? There are good things here. Groups were covered pretty well and I ingested enough material to actually prove a few things independent of the sketch-as-proof solution manual (that gave more direction than the weirdly worded exercise). The coverage of rings was long-winded and shallow. Fields were taught hand-in-hand with rings so it's unclear where ring theory stops and field theory starts.
The biggest problem with this text is that very little of this material feels like it matters outside of intellectual showboating. Thanks to the coverage of symmetry groups, I'm looking at my Rubik's cube collection in a whole new way - and I love it, honestly - but that's it.
I know some loose things abstractly, but the book didn't covey why. My love of history was met with brief biographies on important women and men of Abstract Algebra's history. I can understand Andrew Wiles proof of Fermat's Last Theorem and can read Erno Rubik's work on group theory without being a chump.
But I don't have the capacity or good will to recommend this text or this subject to anyone to doesn't care about those two things.
Oh, or if you need to see lots of curious ways isomorphism is implied between two groups, this will help you out.
There are better, clearer, FREE texts on the internet covering this subject.
Don't waste your money, even if it's a rental.
Note: I didn't read all of the special topics, but those might as well be Appendices given they read so different from the rest of the text
October 31, 2016
what's not to love
May 10, 2024
The previous edition is actually better. Some amount of unnecessary trimming in the 9th edition.
June 29, 2018
Working through the first two parts of the book has opened my mind to the real possibilities of Mathematics. I love that the author takes time to express useful applications of the materials being covered. There were a few places in the book that took some time to get through and were confusing, however; with some additional aid online made it through to rings. Because the book is in multiple parts, I will be taking some time to study a few other topics and the come back to this awesome book. I cannot wait to get back into it.
December 16, 2017
2015 Review:
I did find useful information that helped me in the class, but it was very limited. I feel like the book was good for a few concepts, but for the most part I had to rely on the internet.
2017 Review:
Going through the class and actually reading the book, it is actually a really useful guide. In fact, if this is not your textbook for Abstract Algebra, it should definitely be a companion to your text.
I did find useful information that helped me in the class, but it was very limited. I feel like the book was good for a few concepts, but for the most part I had to rely on the internet.
2017 Review:
Going through the class and actually reading the book, it is actually a really useful guide. In fact, if this is not your textbook for Abstract Algebra, it should definitely be a companion to your text.
March 24, 2018
Nobody explains abstract algebra better then Joseph Gallian. This is an awesome book. I have read many books on abstract algebra and my personal experience is this :
Gallian >> Birkhoff >>>> Dummit >>>> Lang >>>> Others
Gallian >> Birkhoff >>>> Dummit >>>> Lang >>>> Others
March 26, 2025
An acquired taste
February 1, 2013
Herstein for the mathematics undergrad. Gallian for the armchair mathematician.
June 18, 2020
I guess old mathematics is dead.
this kind of study ...
I can't reach to the truth with old mathematics.
hm....
numbers...it's all about comparison. big small and euqal
..
if I ask to mathematician "what is apple?"
they can say its color, amount of sweet, and temperature, shape, sorting ...
but these kind of explanation can tell nothing. more example, what if I ask "what am I?"
they can say korean, male, 174cm, 78kg, O blood, borned in January etc...
but these thing can't explain about me.
maybe in the far future, if we know all of universe. future human being could explain what I am.
I'm sure it never be comparison
this kind of study ...
I can't reach to the truth with old mathematics.
hm....
numbers...it's all about comparison. big small and euqal
..
if I ask to mathematician "what is apple?"
they can say its color, amount of sweet, and temperature, shape, sorting ...
but these kind of explanation can tell nothing. more example, what if I ask "what am I?"
they can say korean, male, 174cm, 78kg, O blood, borned in January etc...
but these thing can't explain about me.
maybe in the far future, if we know all of universe. future human being could explain what I am.
I'm sure it never be comparison
May 7, 2022
Overall a very well written book, as far as math books go. The proofs are written in a fairly straightforward/direct manner, and I felt that the problem sets were fairly straightforward too. I enjoyed the bits of trivia included throughout the book, as I do feel that they fit the theme, particularly given the tendency of modern algebraic theorems and lemmas to find unexpected applications in fields as disparate as Complex Analysis and Encryption. The notation was also very readable compared to what you normally get from the topic.
July 18, 2022
Gallian gives a fair and often detailed overview of this fascinating subject, but some sections are ordered poorly in my opinion, which can lead to readers potentially missing important connections and big pictures. I recommend pairing this with Dummit and Footes abstract algebra for a thorough survey of the field, though one could get away with just studying this book. The examples are often detailed and care is taken to illustrate relevant and sometimes hidden steps. Abstract Algebra is one of my favorite subjects, and this book does a quite a good job at introducing it.
March 9, 2024
I am in an arranged marriage with this book. We have been through estrangements, mostly due to the book being difficult and assuming I should be capable of figuring something out even though it has often not provided sufficient priming. I am not afraid to admit that I have cheated on the book, my affairs have been with Pinter’s “A Book of Abstact Algebra� and Judson’s “Abstact Algebra�. I do think with time, as it happened with Jane and Edward, one can grow to tolerate and perhaps one day even like the book.
June 1, 2021
This textbook has become a classic.
What I really love about this book is the amount of examples and practice problems. Sometimes it is more difficult to work out all the details of some examples than to understand proofs. Which I appreciate, since the author is not trying to spoon-feed the reader with elaborate explanations. This makes the book very good for the self study.
What I really love about this book is the amount of examples and practice problems. Sometimes it is more difficult to work out all the details of some examples than to understand proofs. Which I appreciate, since the author is not trying to spoon-feed the reader with elaborate explanations. This makes the book very good for the self study.
August 22, 2017
good book
September 25, 2018
The best intro to abstract algebra out there. If you don't like this book, then fuck you.
December 5, 2020
Best book
September 6, 2021
A classic. No Dummit and Foote—but sometimes that is a very good thing. Probably my favorite of the undergrad Algebra books.
Read
June 1, 2024Decent book. Wish there was more group actions
May 23, 2025
mehhh kinda overrated...
the one big upside to this book is the size of the problem sets, but the structure and explanations are just not great overall imo
the one big upside to this book is the size of the problem sets, but the structure and explanations are just not great overall imo
April 9, 2022
this is a very nice read, there are clear examples, problems, and bonus sections of relevant history after each chapter.
December 30, 2016
Supposedly this is not a children's book, but it is full of pictures and blurbs about random dead people. Hardly contemporary if you ask me! Not recommended for the serious reader.
March 28, 2008
Worst book ever
September 9, 2012
college text book. what a lousy class
November 7, 2012
the best algebra book
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