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Problems in Group Theory
The most effective way to study any branch of mathematics is to tackle its problems. This wide-ranging anthology offers a straightforward approach, with 431 challenging problems in all phases of group theory, from elementary to the most advanced.
The problems are arranged in eleven subgroups, permutation groups, automorphisms and finitely generated Abelian groups, normal series, commutators and derived series, solvable and nilpotent groups, the group ring and monomial representations, Frattini subgroup, factorization, linear groups, and representations and characters. Each chapter features a preface of pertinent definitions and theorems, and full solutions appear in a separate section.
Most of these problems are derived from research papers published since 1950 (a listing of 102 references is supplied). This compilation makes them readily accessible as a supplement to courses in group theory. The presentation places equal emphasis on techniques and results, encouraging the development of both skill and comprehension.
The problems are arranged in eleven subgroups, permutation groups, automorphisms and finitely generated Abelian groups, normal series, commutators and derived series, solvable and nilpotent groups, the group ring and monomial representations, Frattini subgroup, factorization, linear groups, and representations and characters. Each chapter features a preface of pertinent definitions and theorems, and full solutions appear in a separate section.
Most of these problems are derived from research papers published since 1950 (a listing of 102 references is supplied). This compilation makes them readily accessible as a supplement to courses in group theory. The presentation places equal emphasis on techniques and results, encouraging the development of both skill and comprehension.
- GenresMathematics
192 pages, Paperback
First published June 1, 1973
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Displaying 1 - 2 of 2 reviews
August 13, 2022
It's too hard. But then again the same can probably be said for the entire subject of mathematics. To be sure, many things, like Cantor diagonalization, are rather immediately obvious, but the subject on the whole is exceedingly difficult.
The only saving grace is that it is sort of a compendium of some pretty interesting advanced facts.
There are answers. And I have come to appreciate it more.
I guess for those who are very serious about the subject, it can be considered quite useful.
I was reading it some more, and in one of the solutions it appears the mustard came off the hot dog. But this is to be expected from time to time. Especially in this difficult subject. The author was a Bateman professor at Caltech. And it shows. My point is that on the whole his work is quite impressive.
The presentation of symmetric groups is especially nice. He even has a GTM book on permutation groups, which are subgroups of symmetric groups.
Actually, the whole exposition is pleasing. The author obviously has a great understanding of the subject.
As an example, let me consider one interesting problem in the chapter on automorphisms and finitely generated abelian groups. In it the reader is asked to find a finite group that has a normal subgroup whose automorphism group has larger order than that of the group itself.
While I gave up pretty soon and peeked at the answer, and I largely understand that answer, I do sort of wish there was some elaboration as to why the subgroup is required to be normal.
It figures that the answer provided is a permutation group (happens to be of degree 6).
Actually by Cayley's theorem every group is isomorphic to a permutation group. Interestingly, I didn't see this theorem in the book.
I also wonder if there are many other examples (probable), and whether the example given is of smallest order (also probable, and not that hard to check, since the order is only 24).
Of course, we can conclude that in this example, not every automorphism of the subgroup can be extended to the parent group.
Regardless the book gets points for austere beauty, like the subject itself.
Rings rear their ugly head too: in order to introduce monomial representations he needs to cover the group ring.
The Fitting subgroup is in there too, which I believe is the analog of the Jacobson radical for rings. I get a kick out of the fact that "Fitting" isn't descriptive, though it sounds like it, but rather someone's name.
And there's a chapter on the Frattini.
Later (I think in the last chapter) he covers matrix representations more generally (and characters).
Finally, let me correct my initial comment: it isn't too hard if you know some basic group theory. It just took a little while for me to get going.
The problems and the solutions are elegant and clear. Sometimes one can easily come up with alternative solutions. Try to solve the problems on your own first.
Some of them are quite difficult though. For instance when he provides an example of a 2-generated group with a subgroup that is not finitely generated, he uses a linear group. Of course (if you know some, arguably, not so basic group theory) it can't be abelian. But how to find it!?
One very last thing is that I have a problem with the way Dixon defines the upper central series. For, ostensibly, he only tells us that the quotient of each successive term with the previous term is the center of the factor group of G modulo that previous term. But I don't see from that exactly what "Z sub i" is.
The only saving grace is that it is sort of a compendium of some pretty interesting advanced facts.
There are answers. And I have come to appreciate it more.
I guess for those who are very serious about the subject, it can be considered quite useful.
I was reading it some more, and in one of the solutions it appears the mustard came off the hot dog. But this is to be expected from time to time. Especially in this difficult subject. The author was a Bateman professor at Caltech. And it shows. My point is that on the whole his work is quite impressive.
The presentation of symmetric groups is especially nice. He even has a GTM book on permutation groups, which are subgroups of symmetric groups.
Actually, the whole exposition is pleasing. The author obviously has a great understanding of the subject.
As an example, let me consider one interesting problem in the chapter on automorphisms and finitely generated abelian groups. In it the reader is asked to find a finite group that has a normal subgroup whose automorphism group has larger order than that of the group itself.
While I gave up pretty soon and peeked at the answer, and I largely understand that answer, I do sort of wish there was some elaboration as to why the subgroup is required to be normal.
It figures that the answer provided is a permutation group (happens to be of degree 6).
Actually by Cayley's theorem every group is isomorphic to a permutation group. Interestingly, I didn't see this theorem in the book.
I also wonder if there are many other examples (probable), and whether the example given is of smallest order (also probable, and not that hard to check, since the order is only 24).
Of course, we can conclude that in this example, not every automorphism of the subgroup can be extended to the parent group.
Regardless the book gets points for austere beauty, like the subject itself.
Rings rear their ugly head too: in order to introduce monomial representations he needs to cover the group ring.
The Fitting subgroup is in there too, which I believe is the analog of the Jacobson radical for rings. I get a kick out of the fact that "Fitting" isn't descriptive, though it sounds like it, but rather someone's name.
And there's a chapter on the Frattini.
Later (I think in the last chapter) he covers matrix representations more generally (and characters).
Finally, let me correct my initial comment: it isn't too hard if you know some basic group theory. It just took a little while for me to get going.
The problems and the solutions are elegant and clear. Sometimes one can easily come up with alternative solutions. Try to solve the problems on your own first.
Some of them are quite difficult though. For instance when he provides an example of a 2-generated group with a subgroup that is not finitely generated, he uses a linear group. Of course (if you know some, arguably, not so basic group theory) it can't be abelian. But how to find it!?
One very last thing is that I have a problem with the way Dixon defines the upper central series. For, ostensibly, he only tells us that the quotient of each successive term with the previous term is the center of the factor group of G modulo that previous term. But I don't see from that exactly what "Z sub i" is.
This entire review has been hidden because of spoilers.
January 22, 2015
This very dense book is one that can be used by the experienced group theorist to freshen up their skills for a major examination. It can also serve as a source of solutions or hints to problems that are being given out in a high-level algebra class. This means that instructors can find quality problems to give out to the class and students may be able to find quality hints or even solutions to problems they receive.
The basic format for the eleven chapters is as follows:
*) At most a page of lead in material, it is primarily definitions and explanations of new notation.
*) A set of stated theorems with no proofs given.
*) A set of problems for the reader to work through.
Brief solutions to all of the problems are given in the second section of the book.
This would be a difficult book to use for self study at any point except after the reader has been exposed to group theory starting at a more basic level. For those people beyond that point it is an excellent resource for quality problems.
This review appears on Amazon
The basic format for the eleven chapters is as follows:
*) At most a page of lead in material, it is primarily definitions and explanations of new notation.
*) A set of stated theorems with no proofs given.
*) A set of problems for the reader to work through.
Brief solutions to all of the problems are given in the second section of the book.
This would be a difficult book to use for self study at any point except after the reader has been exposed to group theory starting at a more basic level. For those people beyond that point it is an excellent resource for quality problems.
This review appears on Amazon
Displaying 1 - 2 of 2 reviews