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Conditions for subnormality of a join of subnormal subgroups

Published online by Cambridge University Press:  24 October 2008

J. P. Williams
Affiliation:
Trinity College, Cambridge

Extract

The object of this paper is to prove a necessary and sufficient condition on two groups H, K for their join always to be subnormal in a group G whenever they are embedded subnormally in G.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Fuchs, L.Abelian groups (Pergamon Press, Oxford, 1960).Google Scholar
(2)Fuchs, L.Infinite abelian groups, vol. 1 (Academic Press, New York, 1970).Google Scholar
(3)Gruenberg, K. W.Cohomological topics in group theory (Lecture Notes in Mathematics, no. 143, Springer-Verlag, Berlin, 1970).CrossRefGoogle Scholar
(4)Hall, M.The theory of groups (Macmillan, New York, 1959).Google Scholar
(5)Hall, P.A contribution to the theory of groups of prime-power order. Proc. London Math. Soc. (2) 36 (1934), 2995.CrossRefGoogle Scholar
(6)Lennox, J. C., Segal, D. and Stonehewer, S. E.The lower central series of a join of subnormal subgroups. Math. Z. 154 (1977), 8589.CrossRefGoogle Scholar
(7)Lennox, J. C. and Stonehewer, S. E.The join of two subnormal subgroups. J. London Math. Soc. (2) 22 (1980), 460466.CrossRefGoogle Scholar
(8)Petresco, J.Sur les commutateurs. Math. Z. 61 (1954), 343356.CrossRefGoogle Scholar
(9)Robinson, D. J. S.A property of the lower central series of a group. Math. Z. 107 (1968), 225231.CrossRefGoogle Scholar
(10)Robinson, D. J. S.Infinite soluble and nilpotent groups (Queen Mary College Mathematics Notes, Queen Mary College, London, 1967).Google Scholar
(11)Robinson, D. J. S.Joins of subnormal subgroups. Illinois J. Math. 9 (1965), 144168.CrossRefGoogle Scholar
(12)Roseblade, J. E.The permutability of orthogonal subnormal subgroups. Math. Z. 90 (1965), 365372.CrossRefGoogle Scholar
(13)Roseblade, J. E. and Stonehewer, S. E.Subjunctive and locally coalescent classes of groups. J. Algebra 8 (1968), 423435.CrossRefGoogle Scholar
(14)Stonehewer, S. E.Nilpotent residuals of subnormal subgroups. Math. Z. 139 (1974), 4554.CrossRefGoogle Scholar
(15)Wielandt, H.Topics in the theory of composite groups (University of Wisconsin Lecture Ntoes, University of Wisconsin, Madison, 1967).Google Scholar
(16)Williams, J. P.The join of several subnormal subgroups, Math. Proc. Cambridge Philos. Soc. 92, 391399.CrossRefGoogle Scholar
(17)Zassenhaus, H.The theory of groups, 2nd edition (Chelsea, New York, 1958).Google Scholar