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Endliche p-Gruppen. (Finite p-groups). (German) Zbl 0699.20001

Budapest: Akadémiai Kiadó. 304 S. DM 36.00 (1989).
The main results presented in this book represent the research done by the author during the 1970’s and have not been published in detail previously. He completed it before his death in 1980, and it has been edited by L. Márki and P. P. Pálfy. Finite p-groups are studied by taking a normal series with cyclic factors and writing an element of the group as a product of powers of generators of the factors. The defining relations of the group can thus be written in terms of commutators and powers. One may get such relations as \(x^{-1}yx=y^ a\), and it is when one asks for \(xyx^{-1}\) that one sees the necessity of regarding a as a p-adic integer.
The first chapter thus deals with the field of p-adic numbers and basic p-adic analysis. It starts with the definition of p-adic integers as inverse limits and the proof that they form a local ring. The field of p- adic numbers and divisibility in it are then presented. The convergence of sequences and series are then defined without the use of topological terminology, and the standard digital representation of p-adic numbers follows. The periodicity in this representation of rational numbers is proved, and this is followed by the determination of the roots of unity in the field of p-adic numbers. Continuity and differentiability are studied and the valid analogues of these properties in real variable are proved. After the rather belated definition of the valuation, p-adic power series and their convergence ideals are studied, and this is followed by the p-adic exponential, logarithmic and power functions. The next section is on uniform convergence and ends with Mahler’s theorem that any continuous function on the p-adic integers can be expanded as a uniformly convergent Newton series. The last section of the chapter deals with generalized sums in which the upper limit of summation is a p-adic integer; this is based on a paper by Márki and the author. The whole chapter is a model of mathematical exposition and can be recommended to any student beginning the study of p-adic analysis.
The second chapter, on the author’s “natural theory” of finite p- groups, starts at the same elementary level as the first, and in the beginning it is proved that p-groups have normal series with cyclic factors. It is then shown that if a is an element of a p-group and u is a p-adic integer, then the u-th power of a can be defined. The author states that the ring of p-adic integers thereby becomes a domain of operators for the group, although of course \((ab)^ u=a^ ub^ u\) is not valid in general. By a basis of a p-group is meant the sequence of generators of the factors of a normal series with cyclic factors, here called a fundamental chain, and the terminology and notation in connection with such bases is then set up. Then the “structure constants” are defined: these are the exponents (regarded as p-adic integers) which occur in the standard defining relations arising from a fundamental chain. The associativity condition on the structure constants follows, and then the very complicated functions involved in forming products and powers of elements in their normal form are defined (inductively); as indicated above, the formation of the inverse makes the introduction of p-adic numbers a necessity. A second form of the associativity condition is used to find them explicitly for groups of normal length 3, but it must be said that the calculation here is of a complexity which scarcely permits verification. Some remarks are then made about the shortest length of a fundamental chain, and this section ends with the definition of “parameter classes”. The possibility of “diminishing” a basis by replacing at least one of its elements by its p-th power is then discussed; the most useful bases are thus those which are of minimal length and cannot be diminished in this sense. The following sections are on the more familiar ground of higher commutators and upper and lower central series, though words like “nilpotent” are conspicuous by their absence. But it is proved that the k-th term of the lower central series of a p-group is generated by all the simple commutators of weight k in the elements of a basis. The usual properties of the Frattini subgroup are then established. After a preliminary section on matrices, a division into so-called signature classes is defined, but this is non-trivial only for \(p=2\). Finally, a similar division into defect classes is described.
It will thus be seen that despite its title, the book is not devoted to theories of p-groups previously developed, and indeed, it has little relation to them. What it does set out to do is shown by the application to metacyclic groups in the last chapter. These split into 3 or 7 defect classes according to whether \(p>2\) or \(p=2\). Each of these splits into at most 2 parameter classes, which are precisely described.
Reviewer: N.Blackburn

MSC:

20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20D15 Finite nilpotent groups, \(p\)-groups
20F18 Nilpotent groups
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
20F05 Generators, relations, and presentations of groups
20F12 Commutator calculus
20F14 Derived series, central series, and generalizations for groups
30G06 Non-Archimedean function theory
12J25 Non-Archimedean valued fields
01A75 Collected or selected works; reprintings or translations of classics
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