Abstract
Let G be a finite group and A a subgroup of G. We show that the Kegel–Wielandt conjecture, proved by Peter Kleidman in 1991, implies that A is subnormal in G iff \(\left| AB\right| \) divides \(\left| G\right| \) for all subgroups B of G. We also state and prove a nilpotency criterion for G and a characterization of the Fitting subgroup of G, which are related to this observation.
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References
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I am grateful to Martino Garonzi for a useful discussion.
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Levy, D. The size of a product of two subgroups and subnormality. Arch. Math. 118, 361–364 (2022). https://doi.org/10.1007/s00013-022-01710-8
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DOI: https://doi.org/10.1007/s00013-022-01710-8