Character theory of finite groups.
Corrected reprint of the 1976 original.

*(English)*Zbl 1119.20005
Providence, RI: AMS Chelsea Publishing (ISBN 0-8218-4229-3/hbk). xii, 303 p. (2006).

The book under review is a reprint of the original 1976 publication Character Theory of Finite Groups (Zbl 0337.20005). It was also reprinted as a paperback by Dover Publications in 1994. The reviewer bought a copy of the work in 1978 and is of the opinion that its typography and design are much superior to this current AMS Chelsea version. The new typeface used is too dark (a result of the copying process, which is presumably done by scanning) and the page size too large, leading to excessively wide margins and too much white space. There is also an indistinctness or fuzziness about certain images, such as straight lines. The cost of the new production is certainly less, comparatively speaking, than the former one, and this may be given as a reason for the loss of quality. We noticed one typographical error (dividies in Theorem 6.15), which occurs in the original and reprint copies. The texts are nonetheless remarkably free from typographical error and testify to painstaking proof reading.

Isaacs’s book has established itself as a key text in group character theory, and in many respects it has not been surpassed by any subsequent production. A very full review by Stephen D. Smith appeared in Mathematical Reviews (57, #417), and we refer the interested reader to that source. Isaacs frequently supplied new proofs of existing results, or found unusual applications of character-theoretic principles, and could never be accused of slavishly following the earlier literature. His own research interests probably lay more in chapters 11-13, and he has proceeded to write well over 100 papers, mainly on characters of solvable groups or their generalizations, some of whose origins are arguably traceable back to ideas emerging in his book. Chapter 7, on trivial intersection sets and allied concepts, while containing some lovely applications of fairly deep character theory, may seem less pertinent today. Chapter 6 on the other hand is a thorough introduction to that indispensable tool, Clifford theory, of which Isaacs is an acknowledged master. Perhaps the largest omissions are no mention of some of any of the great families of finite groups, such as the symmetric and alternating groups, or the finite groups of Lie type. These of course have been a major source of problems and new techniques in group representation theory, and the book under review has virtually no intersection with these topics. It is probably of much more assistance to those who wish to study solvable groups, which might have been the author’s unstated intention. Clearly, not all topics can be treated and some personal preference is allowed and expected.

The reviewer found the book gave little idea of how the subject of group character theory emerged as a useful tool for studying groups. Most theorems are quoted without reference to when or how they were proved originally, and one would have little idea that the subject was largely the creation of Frobenius. There is no mention in the references of the collected works of Frobenius, which appeared in 1968 and are informative in fixing chronology and the original working methods. Again, the author probably had no interest in this direction and preferred to concentrate on useful material, not historical derivation. Historical footnotes are more common nowadays, partly as a way of sweetening too much hard reading.

We are sure there is still a sizeable readership for Isaacs’s excellent and innovative text. It is still the best introduction to Clifford theory and its extensions, and its problem sections contain many new and demanding problems, another admirable feature of the work. It is a pity that nobody has undertaken a similar work, filling gaps left by Isaacs, such as block theory, characters of special groups, connections with other parts of mathematics such as number theory, but perhaps that would be a hopeless task.

Isaacs’s book has established itself as a key text in group character theory, and in many respects it has not been surpassed by any subsequent production. A very full review by Stephen D. Smith appeared in Mathematical Reviews (57, #417), and we refer the interested reader to that source. Isaacs frequently supplied new proofs of existing results, or found unusual applications of character-theoretic principles, and could never be accused of slavishly following the earlier literature. His own research interests probably lay more in chapters 11-13, and he has proceeded to write well over 100 papers, mainly on characters of solvable groups or their generalizations, some of whose origins are arguably traceable back to ideas emerging in his book. Chapter 7, on trivial intersection sets and allied concepts, while containing some lovely applications of fairly deep character theory, may seem less pertinent today. Chapter 6 on the other hand is a thorough introduction to that indispensable tool, Clifford theory, of which Isaacs is an acknowledged master. Perhaps the largest omissions are no mention of some of any of the great families of finite groups, such as the symmetric and alternating groups, or the finite groups of Lie type. These of course have been a major source of problems and new techniques in group representation theory, and the book under review has virtually no intersection with these topics. It is probably of much more assistance to those who wish to study solvable groups, which might have been the author’s unstated intention. Clearly, not all topics can be treated and some personal preference is allowed and expected.

The reviewer found the book gave little idea of how the subject of group character theory emerged as a useful tool for studying groups. Most theorems are quoted without reference to when or how they were proved originally, and one would have little idea that the subject was largely the creation of Frobenius. There is no mention in the references of the collected works of Frobenius, which appeared in 1968 and are informative in fixing chronology and the original working methods. Again, the author probably had no interest in this direction and preferred to concentrate on useful material, not historical derivation. Historical footnotes are more common nowadays, partly as a way of sweetening too much hard reading.

We are sure there is still a sizeable readership for Isaacs’s excellent and innovative text. It is still the best introduction to Clifford theory and its extensions, and its problem sections contain many new and demanding problems, another admirable feature of the work. It is a pity that nobody has undertaken a similar work, filling gaps left by Isaacs, such as block theory, characters of special groups, connections with other parts of mathematics such as number theory, but perhaps that would be a hopeless task.

Reviewer: Roderick Gow (Dublin)

##### MSC:

20Cxx | Representation theory of groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |