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Finite p-groups with normal normalisers

  • Elizabeth A. Ormerod (a1) and Gemma Parmeggiani (a2)

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We consider the class N of groups in which the normaliser of every subgroup is normal, and the class C of groups in which the commutator subgroup normalises every subgroup. It is clear that CN, and it is known that groups in the class N are nilpotent of class at most 3. We show that every finite p-group in N is also in C, provided that p ≥ 5, and we give an example showing that this is not true for p = 2.

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References

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[1] Bryce, R.A. and Cossey, J., ‘A note on groups with Hamiltonian quotients’, Rend. Sem. Mat. Univ. Padova 100 1998, 111.
[2] Hobby, C., ‘Finite groups with normal normalizers’, Canad. J. Math. 20 1968, 12561260.
[3] Heineken, H., ‘A class of three-Engel groups’, J. Algebra 17 1971, 341345.
[4] Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967).
[5] Mahdavianary, S.K., ‘A special class of three-Engel groups’, Arch. Math. (Basel) 40 1983, 193199.
[6] Mahdavianary, S.K., ‘A classification of 2-generator p-groups, p ≥ 3, with many subgroup 2-subnormal’, Arch. Math. (Basel) 43 1984, 97107.
[7] Parmeggiani, G., ‘On finite p-groups of odd order with many subgroups 2-subnormal’, Comm. Algebra 24 1996, 27072719.
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