Un livre pour Alain Debreil, alias AD

Trop cher !!
Mais, riche !!

Cordialement,
Yannguyen

" Un coup de dés jamais n'abolira le hasard, Mallarmé "


Yakov Berkovich ; Zvonimir Janko : Groups of Prime Power Order. (De Gruyter Expositions in Mathematics) 1st Edition.


ISBN : 978-3110207170



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Page numbers are just like the physical edition
Length: 639 pages

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TOIC :

This volume constitutes the third part of the impressive contributions on p-groups in recent times by the authors; besides that, in this volume one can find 892 so-called Research Problems and Themes, only a few of them are solved. There are 52 sections here and 18 appendices as well. As there is a wealth of totally new results to mention in this book, we state here only a few.

In this Volume 3 the following problems are solved, all with full proofs,
1)

the classification of p-groups containing only one maximal subgroup which is neither Abelian nor minimal non-Abelian;
2)

the classification of groups all of whose nonnormal subgroups are of equal order;
3)

the classification of p-groups all of whose nonnormal subgroups have normalizers of index p;
4)

the determination of the number of subgroups of given order in metacyclic p-groups;
5)

the computation of the order of the derived group of a group with subgroup breath 1;
6)

the classification of p-groups all of whose subgroups have derived subgroups of order at most p;
7)

the classification of p-groups G all of whose maximal Abelian subgroups containing a non-G-invariant cyclic subgroup of minimal order, say p n , have order at most p n+1 .

The authors give in addition in the Preface in Volume 3 a list of eighteen results proved on finite p-groups not existing in other books devoted to finite p-groups. We mention nine of them here; all eighteen subjects are to be found in Volume 3.
1)

2-groups of sectional rank 4;
2)

p-groups not being generated by certain minimal non-Abelian subgroups;
3)

classification of p-groups with precisely one maximal subgroup neither being Abelian nor minimal non-Abelian;
4)

classification of p-groups all of whose maximal subgroups have cyclic derived subgroups;
5)

p-groups of breadth 2;
6)

groups containing a soft subgroup;
7)

the classification of p-groups all of whose proper subgroups have derived subgroups of order at most p;
8)

the description of all possible sets of numbers of generators of maximal subgroups of 2-generator 2-groups;
9)

a proof of Schenkman’s theorem on the norm of a finite group and a new proof of Baer’s theorem on an arbitrary 2-group with non-Abelian norm.

An essential part in Volume 3 is devoted to the investigation of the influence of minimal non-Abelian subgroups on the structure of a p-group.

As suggested above we were only able to provide the reader of this review with a glimpse at the contents of this Volume 3. The conclusion is that the contents of the three volumes together give an upshot in our knowledge about p-groups. – Highly recommendated all this!

As to the reviews of Volumes 1 and 2 [see Y. Berkovich, Groups of prime power order. Vol. 1. de Gruyter Expositions in Mathematics 46. Berlin: Walter de Gruyter (2008; Zbl 1168.20001) and Y. Berkovich and Z. Janko, Groups of prime power order. Vol. 2. Expositions in Mathematics 47. Berlin: Walter de Gruyter (2008; Zbl 1168.20002)].
Reviewer: R. W. van der Waall (Huizen)