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Groups in which normality is a transitive relation

Published online by Cambridge University Press:  24 October 2008

Derek J. S. Robinson
Derek J. S. Robinson
Affiliation:
Trinity College, Cambridge
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Extract

A group is said to have the property T or to be a T-group if every subnormal subgroup is normal. Thus the class of T-groups is just the class of all groups in which normality is a transitive relation. Finite T-groups have been studied by Best and Taussky (l), Gaschütz (4) and Zacher (11). Gaschütz has shown that if G is a finite soluble T-group and G/L is the unique maximal nilpotent quotient group of G, then G/L is Abelian or Hamiltonian and L is an Abelian group of odd order prime to |G:L| ((4), Satz 1). Our aim is to study infinite T-groups and more especially infinite soluble T-groups with a view to extending Gaschütz's results. One of the simplest results on soluble T-groups is THEOREM 2.3.1. Every soluble T-group is metabelian.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 1 , January 1964 , pp. 21 - 38
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Best, E. and Taussky, O.A class of groups. Proc. Roy. Irish. Acad. Sect. A, 47 (1942), 55–62Google Scholar
(2)Feit, W. and Thompson, J.A solvability criterion for finite groups and some consequences. Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 968–970CrossRefGoogle ScholarPubMed
(3)Fuchs, L.Abelian groups (Pergamon Press, 1960).Google Scholar
(4)Gaschütz, W.Gruppen in denen das Normalteilersein transitiv ist. J. Reine Angew. Math. 198 (1957), 87–92CrossRefGoogle Scholar
(5)Hasse, H.Zahlentheorie (Akademie, 1949).Google Scholar
(6)Kovacs, L. G., Neumann, B. H. and De Vries, H.Some Sylow subgroups. Proc. Roy. Soc. London, Ser. A, 260 (1961), 304–316Google Scholar
(7)Kuroš, A. G.The theory of groups (Chelsea, 1956).Google Scholar
(8)Levi, F. W.Groups in which the commutator operation satisfies certain algebraic conditions. J. Indian Math. Soc. 6 (1942), 87–97Google Scholar
(9)Mclain, D. H.Local theorems in universal algebras. J. London Math. Soc. 34 (1959), 177–184CrossRefGoogle Scholar
(10)Taunt, D. R.Remarks on the isomorphism problem in theories of construction of finite groups. Proc. Cambridge Philos. Soc. 51 (1955), 16–24CrossRefGoogle Scholar
(11)Zacher, G.Caratterizzazione dei t-gruppi finiti risolubili. Ricerche Mat. 1 (1952), 287–294Google Scholar
(12)Zappa, G.Sui gruppi di Hirsch supersolubili I. Rend. Sem. Math. Univ. Padova, 12 (1941), 1–11Google Scholar
(13)Zassenhaus, H.The theory of groups (Chelsea; 2nd ed., 1958).Google Scholar