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Fitting classes of CC-groups

Published online by Cambridge University Press:  13 July 2011

Martyn R. Dixon
Affiliation:
Department of MathematicsUniversity of AlabamaP.O. Box 1416Tuscaloosa, AL 35487, USA
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The theory of Fitting classes is, by now, a well established part of the theory of finite soluble groups. In contrast, Fitting classes have received rather scant attention in infinite groups, although some recent work of Beidleman and Karbe [2] and Beidleman, Karbe and Tomkinson [3] suggest that one can obtain results in this direction. The paper [2], cited above, in fact generalizes earlier work of Tomkinson [9] to the class of locally soluble FC-groups. The present paper is concerned with the theory of Fitting classes in a class of groups somewhat similar to the class of FC-groups, namely the class of CC-groups, introduced by Polovickiǐ in [6]. A group G is a CC-group if G/CG(xG) is a Černikov group for all xG where, as in the rest of this paper, we use the standard group theoretic notation of [7]. Recently, Alcázar and Otal [1] have shown how to generalize results of B. H. Neumann [5] to the class of CC-groups. The main purpose of the present note is to illustrate further how one can handle CC-groups, in an analogous manner to FC-groups, by using techniques similar to those used in [1] and [4].

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

1.Alcazar, J. and Otal, J., Sylow subgroups of groups with Černikov conjugacy classes, J. Algebra, 110 (1987), 507513.CrossRefGoogle Scholar
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3.Beidleman, J. C., Karbe, M. J. and Tomkinson, M. J., Fitting classes of G1groups, I, to appear.Google Scholar
4.Dixon, M. R., Some topological properties of residually Černikov groups, Glasgow Math. J. 23 (1982), 6582.CrossRefGoogle Scholar
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7.Robinson, D. J. S., Finiteness Conditions and Generalized Soluble Groups (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
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9.Tomkinson, M. J., ℱ-injectors of locally soluble FC-groups, Glasgow Math. J. 10 (1969), 130136.CrossRefGoogle Scholar
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