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Description of finite nilpotent groups of 2nd degree with prime odd period. (English. Russian original) Zbl 1188.20012

Russ. Math. 52, No. 12, 13-22 (2008); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2008, No. 12, 17-27 (2008).
From the introduction: Finite \(p\)-groups are nilpotent. Below we consider only groups whose period \(p\) is a prime odd number and whose nilpotency degree equals 2. Groups with 2-generated centralizers are described by the author and by O. O. Mazurok [in Ukr. Math. J. 50, No. 4, 605-611 (1998); translation from Ukr. Mat. Zh. 50, No. 4, 534-539 (1998; Zbl 0940.20039)]. Below we adduce an algorithm which describes the mentioned groups with an arbitrary centralizer, enumerating the values of commutators of generating elements of the groups. In addition, these groups are defined by operations on tuples of scalars from a Galois field with prime odd characteristic. We consider odules over the Galois field on the studied groups, thus we also describe these odules. The latter are used for the construction of finite planes. The use of odules enables one to prove that finite nilpotent groups of degree 2 with \(m\) generating elements are groups with unique roots.

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20F05 Generators, relations, and presentations of groups
20F12 Commutator calculus

Citations:

Zbl 0940.20039
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References:

[1] A. I. Dolgarev, ”To the Problem of Description of Finite Metabelian Groups of Degree p,” in IX Vsesouzn. Algebraicheskii Kollokvium (Gomel, 1968), p. 69.
[2] O. O. Mazurok, ”Groups with the Elementary Abelian Centralizer of the Order Not Greater Than p 2,” Ukr. Matem. Zhurn. 50(4), 534–539 (1998). · Zbl 0940.20039
[3] A. I. Dolgarev, ”Finite Odular Planes and Modular Trellises,” in Mezhdunar. Konf. po Algebre Pamyati A. I. Shirshova, Tez. Dokl. po Logike i Univers. Algebram, Prikl. Algebre (Novosibirsk, 1991), p. 38.
[4] L. V. Sabinin, ”Odules as a New Approach to the Connected Geometry,” Sov. Phys. Dokl. 5, 800–803 (1977).
[5] L. A. Skornyakov (Ed.), General Algebra (Nauka, Moscow, 1990), Vol. 1 [in Russian].
[6] M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory (Nauka, Moscow, 1982) [in Russian]. · Zbl 0884.20001
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