Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T23:17:30.427Z Has data issue: false hasContentIssue false
  • English
  • Français

On ℱ-residuals of finite groups

Published online by Cambridge University Press:  17 April 2009

Wenbin Guo ,
K. P. Shum  and
A. N. Skiba
Wenbin Guo
Affiliation:
Department of Mathematics, Xuzhou Normal University, Xuzhou, 221009People's Republic of China e-mail: [email protected]
K. P. Shum
Affiliation:
Department of Mathematics, The Chinese University of Hong KongShatin, Hong KongPeople's Republic of China (SAR), e-mail: [email protected]
A. N. Skiba
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, People's Republic of China (SAR) e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that there exists a soluble, saturated and s-closed formation ℱ of groups such that the class (Gℱ | G is a group, Gℱ is the ℱ-residual of G) is not closed under subdirect products. This result a negative answer to an open problems recently proposed by L.A. Shemetkov in 1998.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 2 , April 2002 , pp. 271 - 275
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Doerk, K. and Hawkes, T., ‘On the residual of a direct product’, Arch. Math. (Basel) 30 (1978), 458–468.CrossRefGoogle Scholar
[2]Doerk, K. and Hawkes, T., Finite soluble groups, de Gruyter Expositions in Mathematics 4 (Walter de Gruyter, Berlin, New York, 1992).CrossRefGoogle Scholar
[3]Guo, W., The theory of classes of groups (Science Press-Kluwer Academic Publishers, Beijing, New York, Dordrecht, Boston, London, 2000).Google Scholar
[4]Robinson, D.J.S., A course in the theory of groups, Graduate Texts in Mathematics 80 (Springer-Verlag, New York, Heidelberg, Berlin, 1982).CrossRefGoogle Scholar
[5]Shemetkov, L.A., ‘Gaschütz products of group classes’, Dokl. Akad. Nauk Belarusi 42 (1998), 22–26.Google Scholar
[6]Willems, W., ‘On the projectives of a group algebra’, Math. Z. 171 (1980), 163–174.CrossRefGoogle Scholar