On groups related to
the Čarin groups

O.D. Artemovych

We establish some properties of groups $G$ with the hypercentral commutator subgroup $G'$ which is a $p'$-group and the divisible $p$-Černikov quotient group $G/G'$, and construct several examples of those groups.

1. V. S. Čarin, A remark on the minimal condition for subgroups // Dokl. Akad. Nauk USSR V. 66 – 1949 p. 575–576 Russian

2. V. V. Belyaev, Locally finite groups with almost abelian proper subgroups // Siberian Math. Zh. V. 24 – 1983 p. 11–17 Russian

3. H. Heineken, I. J. Mohamed A group with trivial centre satisfying the normalizer condition // J. Algebra V. 10 – 1968 p. 368–376

4. J. D. P. Meldrum, On the Heineken-Mohamed groups // J. Algebra V. 27 – 1973 p. 437–444

5. B. Hartley, A note on the normalizer condition // Proc. Cambr. Phil. Soc. V. 74 – 1973 p. 11–15

6. B. Bruno, R. E. Phillips, On multipliers of Heineken-Mohamed type groups // Rend. Sem. Mat. Padova V. 85 – 1991 p. 133–146

7. F. Menegazzo, Groups of Heineken-Mohamed // J. Algebra V. 171 – 1995 p. 807–825

8. B. Bruno, On groups with abelian-by-finite proper subgroups // Boll. Un. Mat. Ital. V. (6)3-B – 1984 p. 797–807

9. B. Bruno, Special $q$-groups and ${\Bbb C}_{p^\infty}$-groups of automorphisms // Archiv Math. V. 48 -- 1987 p. 15--24

10. B. Bruno, On $p$-groups with "nilpotent-by-finite" proper subgroups // Bull. Un. Mat. Ital. V. (7)3-A -- 1989 p. 45--51

11. B. Bruno, R. E. Phillips, A note on groups with nilpotent-by-finite proper subgroups // Archiv Math. V. 65 – 1995 p. 369–374

12. O. D. Artemovych, Indecomposable metabelian groups // Ukrainian Math. Zh. V. 42 – 1990 p. 1252–1254 Russian

13. O. D. Artemovych, On groups with hypercentral-by-finite proper subgroups // Dopovidi NANU – 1997 V. 8 p. 7–9 Ukrainian

14. A. O. Azar, On nonnilpotent $p$-groups and the normalizer condition // Turkish J. Math. V. 18 -- 1994 p. 114--129

15. D. J. S. Robinson, A course in the theory of groups , Springer ,addr New York etc. – 1980

16. C. Faith, Algebra: rings, modules and categories Vol.I. , Springer ,addr Berlin e.a. – 1973

17. C. Faith, Algebra Vol.II.. Ring theory , Springer ,addr Berlin e.a. – 1976

18. O. D. Artemovych, On Frobenius groups associated with modules // Demonstratio Math. – 1998 V. 31 № 4 p. 875–878

19. Y. P. Sysak, Products of infinite abalian groups info Preprint 82.59 , Institute of Maths ,addr Kiev – 1982

20. B. Amberg, S. Franciosi, F. de Giovanni, Products of groups , Oxford Mathematical Monographs, Clarendon Press – 1992

21. C. J. Maxson, Rigid rings // Pros. Edinburgh Math. Soc – 1979 V. 21 № 2 p. 95–101

22. J. Watterts, On the adjoint group of a radical ring // J. London Math. Soc – 1968 V. 43 p. 725–729

23. B. Amberg, O. Dickenschied, On the adjoint group of a radical ring // Canad. Math. Bull – 1995 V. 38 № 3 p. 262–270

24. R. F. Faudree, Locally finite and solvable subgroups of sfields // Proc. Amer. Math. Soc – 1969 V. 22 p. 407–417

25. O. D. Artemovych, Yu. B. Ishchuk, On semiperfect rings determined by the adjoint group // Math. Studii – 1997 V. 8 № 2 p. 162–170

26. A. W. Chatters, C. R. Hajarnavis, Rings with chain conditions , Pitman ,addr San Francisco a.o. – 1980

27. D. S. Passman, The algebraic structure of group rings , J.Wiley-Interscione ,addr New York – 1977

	On groups related to the~Čarin groups
Author	O.D. Artemovych topos@prima.franko.lviv.ua Department of Algebra and Mathematical Logic, Faculty of Mathematics and MechanicsKyiv Taras Shevchenko University, Volodymyrs'ka St 64252033, Kyiv, Ukraine
Abstract	We establish some properties of groups $G$ with the hypercentral commutator subgroup $G'$ which is a $p'$-group and the divisible $p$-Černikov quotient group $G/G'$, and construct several examples of those groups.
Keywords	groups with hypercentral commutator subgroup, ivisible p-Černikov quotient, group properties, group theory, examples of groups
DOI	doi:10.30970/ms.11.2.141-148
Reference	1. V. S. Čarin, A remark on the minimal condition for subgroups // Dokl. Akad. Nauk USSR V. 66 – 1949 p. 575–576 Russian 2. V. V. Belyaev, Locally finite groups with almost abelian proper subgroups // Siberian Math. Zh. V. 24 – 1983 p. 11–17 Russian 3. H. Heineken, I. J. Mohamed A group with trivial centre satisfying the normalizer condition // J. Algebra V. 10 – 1968 p. 368–376 4. J. D. P. Meldrum, On the Heineken-Mohamed groups // J. Algebra V. 27 – 1973 p. 437–444 5. B. Hartley, A note on the normalizer condition // Proc. Cambr. Phil. Soc. V. 74 – 1973 p. 11–15 6. B. Bruno, R. E. Phillips, On multipliers of Heineken-Mohamed type groups // Rend. Sem. Mat. Padova V. 85 – 1991 p. 133–146 7. F. Menegazzo, Groups of Heineken-Mohamed // J. Algebra V. 171 – 1995 p. 807–825 8. B. Bruno, On groups with abelian-by-finite proper subgroups // Boll. Un. Mat. Ital. V. (6)3-B – 1984 p. 797–807 9. B. Bruno, Special $q$-groups and ${\Bbb C}_{p^\infty}$-groups of automorphisms // Archiv Math. V. 48 -- 1987 p. 15--24 10. B. Bruno, On $p$-groups with "nilpotent-by-finite" proper subgroups // Bull. Un. Mat. Ital. V. (7)3-A -- 1989 p. 45--51 11. B. Bruno, R. E. Phillips, A note on groups with nilpotent-by-finite proper subgroups // Archiv Math. V. 65 – 1995 p. 369–374 12. O. D. Artemovych, Indecomposable metabelian groups // Ukrainian Math. Zh. V. 42 – 1990 p. 1252–1254 Russian 13. O. D. Artemovych, On groups with hypercentral-by-finite proper subgroups // Dopovidi NANU – 1997 V. 8 p. 7–9 Ukrainian 14. A. O. Azar, On nonnilpotent $p$-groups and the normalizer condition // Turkish J. Math. V. 18 -- 1994 p. 114--129 15. D. J. S. Robinson, A course in the theory of groups , Springer ,addr New York etc. – 1980 16. C. Faith, Algebra: rings, modules and categories Vol.I. , Springer ,addr Berlin e.a. – 1973 17. C. Faith, Algebra Vol.II.. Ring theory , Springer ,addr Berlin e.a. – 1976 18. O. D. Artemovych, On Frobenius groups associated with modules // Demonstratio Math. – 1998 V. 31 № 4 p. 875–878 19. Y. P. Sysak, Products of infinite abalian groups info Preprint 82.59 , Institute of Maths ,addr Kiev – 1982 20. B. Amberg, S. Franciosi, F. de Giovanni, Products of groups , Oxford Mathematical Monographs, Clarendon Press – 1992 21. C. J. Maxson, Rigid rings // Pros. Edinburgh Math. Soc – 1979 V. 21 № 2 p. 95–101 22. J. Watterts, On the adjoint group of a radical ring // J. London Math. Soc – 1968 V. 43 p. 725–729 23. B. Amberg, O. Dickenschied, On the adjoint group of a radical ring // Canad. Math. Bull – 1995 V. 38 № 3 p. 262–270 24. R. F. Faudree, Locally finite and solvable subgroups of sfields // Proc. Amer. Math. Soc – 1969 V. 22 p. 407–417 25. O. D. Artemovych, Yu. B. Ishchuk, On semiperfect rings determined by the adjoint group // Math. Studii – 1997 V. 8 № 2 p. 162–170 26. A. W. Chatters, C. R. Hajarnavis, Rings with chain conditions , Pitman ,addr San Francisco a.o. – 1980 27. D. S. Passman, The algebraic structure of group rings , J.Wiley-Interscione ,addr New York – 1977
Pages	141-148
Volume	11
Issue	2
Year	1999
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On groups related to the~Čarin groups