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On generalized Chernikov groups

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Senashov V.I.

{\sc Definition}. {\it A periodic almost locally solvable group with a primary

minimality condition is said to be a generalized Cher\-ni\-kov group.}

The term "generalized Cher\-ni\-kov groups" was first developed in [1].

Its use can be justified by the fact that according

to the Theorem of Ya.D. Polovitskii [2], the generalized

Cher\-ni\-kov group $G$ is an extension of the direct product

$A$ of quasicyclic $p$-groups with a finite number of multipliers

for any prime number $p$ by the locally normal group $B$, and each

of the $G$ elements are element-wise nonpermutable only with the finite

number of Sylow primary subgroups from $A$. For a comparison,

the Cher\-ni\-kov group is a finite extension of the direct product of

quasicyclic groups taken in the finite number.

Now we proved the next theorem which generalizes the results from [3,4].

{\sc Theorem.} {\it Let $G$ be a group without involutions. Then $G$

has a generalized Cher\-ni\-kov periodic part if and only if it is

conjugately biprimitively finite and the normalizers of any finite

non-trivial subgroup has a generalized Cher\-ni\-kov periodic part.}

Then $G$ has a generalized Cher\-ni\-kov part

Examples by Novikov, Adjan and Ol'shanskii [5,6] prove that in our

theorem the condition that $G$ is conjugately biprimitively finite cannot be

removed.

\centerline{ References }

1. Shunkov V.P., Shafiro A.A. On one characterization

of general Chernikov groups // Materials of the 15 All Union

algebraic conf. -- Krasnoyarsk: Krasnoyarsk State University, 1979. --

P.~185.

2. Polovitskii Ya.D. Layer-Extremal Groups // Mat.Sbornik.

-- 1962. -- V. 56, No.~1. -- P.~95--106.

3. Senashov V.I. Groups with minimality condition // Proceedings of the

International Conference "Infinite Groups 1994",

held in Ravello, Italy, May 23-27, 1994 / ed Francesco de Giovanny and

Martin L. Newell. --- Berlin; New York: de Gruyter, 1995 --- P. 229---234.

4. Senashov V.I. Characterization of generalized Chernikov groups //

Doclady RAN. - 1997.- Vol 352, No 3. --- P. 309-310.

5. Adian S.I. Burnsaide Problem and Identities in Groups. - Moscow:

Nauka, 1975.- 336 p. (In Russian)

6. Ol'shanskii A.Yu. Geometry of Definite Relations in Groups.

- Moscow: Nauka, 1989.- 448 p. (In Russian)