Chernikov group

Černikov group

A group satisfying the minimum condition on subgroups and having a normal Abelian subgroup of finite index (cf. Artinian group; Abelian group; Group with the minimum condition). Such groups have also been called extremal groups. The structure of Abelian groups with the minimum condition was obtained by A.G. Kurosh (see [a4]), who showed that these are precisely the groups that are the direct sum of finitely many quasi-cyclic groups and cyclic groups of prime-power order (cf. Quasi-cyclic group; Group of type $p^\infty$). A quasi-cyclic group (or Prüfer group of type $p^\infty$, for some fixed prime number $p$) is the multiplicative group of complex numbers consisting of all $p^n$-th roots of unity as $n$ runs through the set of natural numbers. It is clear that subgroups and homomorphic images of Chernikov groups are also Chernikov; further, an extension of a Chernikov group by a Chernikov group is again Chernikov.

Chernikov groups are named in honour of S.N. Chernikov, who made an extensive study of groups with the minimum condition. For example, he showed [a1] that a solvable group with the minimum condition on subgroups is (in contemporary terminology) a Chernikov group. Groups with the minimum condition are periodic (cf. Periodic group). In 1970, V.P. Shunkov [a5] proved that a locally finite group $G$ with the minimum condition is Chernikov, a result which had been conjectured for many years. In fact, Shunkov's result is stronger: he showed in [a6] that to force the locally finite group $G$ to be Chernikov one only needs the condition that all the Abelian subgroups of $G$ have the minimum condition. The first examples of groups with the minimum condition which are not Chernikov were provided in 1979 by A.Yu. Ol'shanskii [a3] and E. Rips. These examples are two-generator infinite simple groups in which every proper subgroup is of prime order $p$.

Chernikov groups have played an important role in the theory of infinite groups. For example, Chernikov proved that a periodic group of automorphisms of a Chernikov group is also Chernikov (see [a2], 1.F.3) and this fact is used on numerous occasions in the theory of locally finite groups. Many characterizations of Chernikov groups have been obtained. For example, a hypercentral group is a Chernikov group if and only if each upper central factor satisfies the minimum condition (see [a4], Thm. 10.23, Cor. 2).

References

[a1]	S.N. Černikov, "Infinite locally soluble groups" Mat. Sb. , 7 (1940) pp. 35–64
[a2]	O.H. Kegel, B.A.F. Wehrfritz, "Locally finite groups" , North-Holland (1973)
[a3]	A.Yu. Ol'šanskii, "Infinite groups with cyclic subgroups" Dokl. Akad. Nauk SSSR , 245 (1979) pp. 785–787 (In Russian)
[a4]	D.J.S. Robinson, "Finiteness conditions and generalized soluble groups 1–2" , Ergebn. Math. Grenzgeb. , 62/3 , Springer (1972)
[a5]	V.P. Šunkov, "On the minimality problem for locally finite groups" Algebra and Logic , 9 (1970) pp. 137–151 Algebra i Logika , 9 (1970) pp. 220–248
[a6]	V.P. Šunkov, "On locally finite groups with a minimality condition for abelian subgroups" Algebra and Logic , 9 (1970) pp. 350–370 Algebra i Logika , 9 (1970) pp. 579–615