On classification of nonisomorphic Chernikov groups
by
Igor Shapochka
Uzhgorod University, Uzhgorod, Ukraine

In [1, 2] have been shown by the methods of the theory of integral representations of finite groups, that the description of all nonisomorphic extensions of an arbitrary divisible abelian p-group with the minimality condition by the finite p-group H is wild problem, if one of the following conditions holds: 1) H is a noncyclic p-group and p =/= 2; 2) H is a noncyclic 2-group of order |H| > 4; 3) H is a cyclic p-group of order p^s (s > 2 if p =/= 2, s > 3 if p=2). All such nonisomorphic extensions were described there in the case, if H is cyclic p-group of order p^s (s <= 2).

We have proved the next theorem.

Theorem. Let H be a finite group with normal Sylow p-subgroup P, which is one of the following 1) P is a noncyclic p-group and p =/= 2; 2) P is a noncyclic 2-group of order |P| > 4; 3) P is a cyclic p-group of order p^s (s > 2 if p =/= 2, s > 3 if p=2). The description of all nonisomorphic extensions of an arbitrary divisible abelian p-group with the minimality condition by the group H is wild problem.

References

[1] P.M. Gudivok, F.G. Vaschuk, V.S. Drobotenko, Chernikov p-groups and integral p-adic representations of finite groups, Ukr. mat. zhurn. 44 (1992), 742-753.

[2] P.M. Gudivok, I.V. Shapochka, On Chernikov p-groups, Ukr. mat. zhurn. 51 (1999), 291-304.

Date received: July 18, 2000