Character theory of finite groups.

*(English)*Zbl 0932.20007
de Gruyter Expositions in Mathematics. 25. Berlin: Walter de Gruyter. vi, 618 p. (1998).

In Bull. Am. Math. Soc. 36(1999), pp. 489-492, I. M. Isaacs has given a referat on this book of Huppert’s. He welcomes it as a very good addition to the existing literature on character theory. Huppert himself mentions that his book can be seen as an addition to the (well-known) book of I. M. Isaacs, published in 1976 under the same title (Zbl 0337.20005), and also to the book of O. Manz and Th. Wolf [Representations of solvable groups (1993; Zbl 0928.20008)]. In Isaacs’ referat a wealth of features of the book has been mentioned; therefore this reviewer sees it now as his task to bring up some new items about the contents of the book.

To the reviewer the appearance of the book came as a total surprise, certainly due to the fact that Huppert published already his famous triology around the years 1967 and 1982, and because other people, like Isaacs, Wolf and Manz, Collins (to mention a few) dealt with the subject of characters of finite groups. Don’t forget also Karpilovsky’s mega-work. Thus why publish a(nother) book on character theory?

In the years 1973 and 1974 Huppert gave a series of lectures on the subject for students in Mainz. The contents were bundeled in an unofficial edition, in three small volumes, by the Mainzer Mathematics Institute. There was certainly a strong wish from group theorists in making these notes more public. It never happened that way. The reviewer could not find in the book under review a reference to those “little 3-fold blue books”, but he assumes that this book is a (final?) compliance to the wish mentioned above. Of course things have been brought up to date and also added. Indeed, in the last two or three decades character theory has started to blossom intensively by work of several representation theorists. Let us look at the contents.

There are 46 so-called sections. About 18 of them are introductory but very eluminating and complete (for instance, groups of order \(p^aq^b\), Frobenius groups, induced characters, Glauberman’s character correspondence). Then Clifford theory and projective representations follow. Starting around section 23 and more or less up to the end, recent resesarch results are given (many with proofs), and also theoretical innovations like \(\pi\)-character theory, degree problems etc. A particular thing, unusual in other textbooks, is a list of “examples of groups”; also the many so-called Remarks are very suited for a more detailed study by the reader.

Of course an author makes his own choice in presenting the material. In comparison to the unofficial 1973-1974 notes the reviewer feels however that at some places a few pages with Remarks on research-results of the last 25 years or so, might have been added; from instance in section 24 on \(M\)-groups.

On the whole it is a good treatise and as suggested implicitly before, the reader should consult the Referat of Isaacs in BAMS.

The printing by the de Gruyter Verlag is excellent and beautiful, as always. The proof-reading seems to be o.k., be it that there are some eye-catching mistakes in the spelling and registration of names (on page 2: Gagula must be Gagola; on page 609 (five times) Glaubermann must be Glauberman; the first name of the reviewer (Robert) has not been printed in its abbreviated form R. on page 614 (twice) and on page 616). These (and other?) inconveniences will be not existent anymore in an eventual next printing of the book.

To close with, the reviewer found this “big blue” Huppert book a welcome enrichment of earlier books on character theory of finite groups.

To the reviewer the appearance of the book came as a total surprise, certainly due to the fact that Huppert published already his famous triology around the years 1967 and 1982, and because other people, like Isaacs, Wolf and Manz, Collins (to mention a few) dealt with the subject of characters of finite groups. Don’t forget also Karpilovsky’s mega-work. Thus why publish a(nother) book on character theory?

In the years 1973 and 1974 Huppert gave a series of lectures on the subject for students in Mainz. The contents were bundeled in an unofficial edition, in three small volumes, by the Mainzer Mathematics Institute. There was certainly a strong wish from group theorists in making these notes more public. It never happened that way. The reviewer could not find in the book under review a reference to those “little 3-fold blue books”, but he assumes that this book is a (final?) compliance to the wish mentioned above. Of course things have been brought up to date and also added. Indeed, in the last two or three decades character theory has started to blossom intensively by work of several representation theorists. Let us look at the contents.

There are 46 so-called sections. About 18 of them are introductory but very eluminating and complete (for instance, groups of order \(p^aq^b\), Frobenius groups, induced characters, Glauberman’s character correspondence). Then Clifford theory and projective representations follow. Starting around section 23 and more or less up to the end, recent resesarch results are given (many with proofs), and also theoretical innovations like \(\pi\)-character theory, degree problems etc. A particular thing, unusual in other textbooks, is a list of “examples of groups”; also the many so-called Remarks are very suited for a more detailed study by the reader.

Of course an author makes his own choice in presenting the material. In comparison to the unofficial 1973-1974 notes the reviewer feels however that at some places a few pages with Remarks on research-results of the last 25 years or so, might have been added; from instance in section 24 on \(M\)-groups.

On the whole it is a good treatise and as suggested implicitly before, the reader should consult the Referat of Isaacs in BAMS.

The printing by the de Gruyter Verlag is excellent and beautiful, as always. The proof-reading seems to be o.k., be it that there are some eye-catching mistakes in the spelling and registration of names (on page 2: Gagula must be Gagola; on page 609 (five times) Glaubermann must be Glauberman; the first name of the reviewer (Robert) has not been printed in its abbreviated form R. on page 614 (twice) and on page 616). These (and other?) inconveniences will be not existent anymore in an eventual next printing of the book.

To close with, the reviewer found this “big blue” Huppert book a welcome enrichment of earlier books on character theory of finite groups.

Reviewer: R.W.van der Waall (Amsterdam)

##### MSC:

20C15 | Ordinary representations and characters |

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

20Cxx | Representation theory of groups |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |