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Prefrattini Subgroups and Cover-Avoidance Properties in π”˜-Groups

  • M. J. Tomkinson (a1)

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W. Gaschutz [5] introduced a conjugacy class of subgroups of a finite soluble group called the prefrattini subgroups. These subgroups have the property that they avoid the complemented chief factors of G and cover the rest. Subsequently, these results were generalized by Hawkes [12], Makan [14; 15] and Chambers [2]. Hawkes [12] and Makan [14] obtained conjugacy classes of subgroups which avoid certain complemented chief factors associated with a saturated formation or a Fischer class. Makan [15] and Chambers [2] showed that if W, D and V are the prefrattini subgroup, 𝔍-normalizer and a strongly pronormal subgroup associated with a Sylow basis S, then any two of W, D and V permute and the products and intersections of these subgroups have an explicit cover-avoidance property.

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References

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1. Birkhoff, G., Lattice theory, Second Ed. (Amer. Math. Soc, Providence, R.I., 1964).
2. Chambers, G. A., On f-Prefrattini subgroups, Can. Math. Bull. 15 (1972), 345–348.
3. Gardiner, A. D., Hartley, B., and Tomkinson, M. J., Saturated formations and Sylow structure in locally finite groups, J. Algebra 17 (1971), 177–211.
4. Gaschutz, W., Uber die S-Untergruppe endlicher Gruppen, Math. Z. 56 (1952), 376–387.
5. Gaschutz, W., Praefrattinigruppen, Arch. Math. 13 (1962), 418–426.
6. Graddon, C. J., Some generalizations, to certain locally finite groups, of theorems due to Chambers and Rose, Illinois J. Math. 17 (1973), 666–679.
7. Hartley, B., C-abnormal subgroups of certain locally finite groups, Proc. London Math. Soc. 23 (1971), 228–258.
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10. Hartley, B., A class of modules over a locally finite group, I. J. Austral. Math. Soc. 16 (1973), 431–442. H# Aclass 0f modules over a locally finite group III, to appear.
12. Hawkes, T. o., Analogues of Prefrattini subgroups, Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ. Canberra, August 1965, pp. 145–150 (Gordon and Breach, New York, 1967).
13. A note on system normalizers of a finite soluble group, Proc. Cambridge Philos. Soc. 62 (1966), 339–346.
14. Makan, A. R., Another characteristic conjugacy class of subgroups of finite soluble groups, J. Austral. Math. Soc. 11 (1970), 395–400.
15. Makan, A. R., On certain sublattices of the lattice of subgroups generated by the Prefrattini subgroups, the injectors and the formation subgroups, Can. J. Math. 25 (1973), 862–869.
16. Tomkinson, M. J., Formations of locally soluble FC-groups, Proc. London Math. Soc. 19 (1969), 675–708.
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Prefrattini Subgroups and Cover-Avoidance Properties in π”˜-Groups

  • M. J. Tomkinson (a1)

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