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On the Wielandt subgroup of infinite soluble groups

  • Rolf Brandl (a1), Silvana Franciosi (a2) and Francesco de Giovanni (a3)

Extract

The Wielandt subgroup w(G) of a group G is defined to be the intersection of the normalizers of all the subnormal subgroups of G. If G is a group satisfying the minimal condition on subnormal subgroups then Wielandt [10] showed that w(G) contains every minimal normal subgroup of G, and so contains the socle of G, and, later, Robinson [6] and Roseblade [9] proved that w(G) has finite index in G.

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References

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1.Baer, R., Kern, Der, eine charakteristische Untergruppe, Compositio Math. 1 (1934), 254283.
2.Cooper, C. D. H., Power automorphisms of a group, Math. Z. 107 (1968), 335356.
3.Cossey, J., The Wielandt subgroup of a polycyclic group, Australian National University, Math. Research Report Series 19 (1988).
4.Hall, P., On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595622.
5.Robinson, D. J. S., Groups in which normality is a transitive relation, Proc. Cambridge Philos. Soc. 60 (1964), 2138.
6.Robinson, D. J. S., On the theory of subnormal subgroups, Math. Z. 89 (1965), 3051.
7.Robinson, D. J. S., A note on groups of finite rank, Compositio Math. 21 (1969), 240246.
8.Robinson, D. J. S., Finiteness conditions and generalized soluble groups (Springer, 1972).
9.Roseblade, J. E., On certain subnormal coalition classes, J. Algebra 1 (1964), 132138.
10.Wielandt, H., Uber den Normalisator der subnormalen Untergruppen, Math. Z. 69 (1958), 463465.

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