On generalized Chernikov groups

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)