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A Class of infinite soluble groups with an A-group condition

Published online by Cambridge University Press:  18 May 2009

M. J. Tomkinson
M. J. Tomkinson
Affiliation:
University of Glasgow, Department of Mathematics, University Gardens, Glasgow, Scotland G12 8QW
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Finite soluble groups in which all the Sylow subgroups are abelian were first investigated by Taunt [8] who referred to them as A-groups. Locally finite groups with the same property have been considered by Graddon [2]. By the use of Sylow theorems it is clear that every section (homomorphic image of a subgroup) of an A-group is also an A-group and hence every nilpotent section of an A-group is abelian. This is the characterization that we use here in considering groups which are not, in general, periodic.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 21 , Issue 1 , January 1980 , pp. 81 - 84
Copyright
Copyright © Glasgow Mathematical Journal Trust 1980

References

REFERENCES

1.Carter, R. W. and Hawkes, T. O., The -normaKzers of a finite soluble group, j. Algebra 5 (1967), 175–202.CrossRefGoogle Scholar
2.Graddon, C. J., Some generalizations, to certain locally finite groups, of theorems due to Chambers and Rose, Illinois J. Math. 17 (1973), 666–679.CrossRefGoogle Scholar
3.Hartley, B. and Tomkinson, M. J., Splitting over nilpotent and hypercentral residuals, Math, Proc. Cambridge Philos. Soc. 78 (1975), 215–226.CrossRefGoogle Scholar
4.Robinson, D. J. S., Finiteness conditions and generalized soluble groups, Part 1 (Springer, 1972).CrossRefGoogle Scholar
5.Robinson, D. J. S., Finiteness conditions and generalized soluble groups, Part 2 (Springer 1972).CrossRefGoogle Scholar
6.Seitz, G. M. and Wright, C. R. B., On complements of -residuals in finite soluble groups Arch. Math. (Basel) 21 (1970), 139–150.CrossRefGoogle Scholar
7.Smel'kin, A. L., Polycyclic groups, Sibirsk. Mat. Ž 9 (1968), 234–235.Google Scholar
8.Taunt, D., On A-groups, Proc. Cambridge Philos. Soc. 45 (1949), 24–42.CrossRefGoogle Scholar
9.Tomkinson, M. J., Formation theory and groups of automorphisms of -groups, Proc. Roy Soc. Edinburgh A 76 (1977), 255–265.CrossRefGoogle Scholar
10.Tomkinson, M. J., Splitting theorems in abelian-by-hypercyclic groups, J. Austral. Math Soc. Ser A 25 (1978), 71–91.CrossRefGoogle Scholar