Wielandt subgroups of nilpotent residuals and Fitting-height bounds

Abstract

For a finite group G, let W(G) be the intersection of the normalizers of the nilpotent residuals of the subnormal subgroups of G, and let ℓ_W(G) be the length of the associated upper series. Thus W(G)=G exactly when all these residuals are normal in G. We prove that every finite soluble group satisfies ℓ_W(G)≤ h(G)≤4ℓ_W(G), where h(G) is the Fitting height, and that h(G)≤3ℓ_W(G) if |G| is odd or the Sylow $2$-subgroups of G are abelian. Both constants are attained at every length, the constant $4$ by {2,3}-groups. Moreover, every pair (n,h) with n≤ h≤4n occurs as (ℓ_W(G),h(G)). The constant $4$ comes from primitive groups. A soluble primitive group Vrtimes M satisfies W(Vrtimes M)=Vrtimes M if and only if every abelian subgroup of F(M) is cyclic. It then has Fitting height four if and only if F(M)=C× Q₈, with C cyclic, and M induces the full group Out(Q₈)≅ S₃. We also determine when the base group of a soluble regular wreath product is the first term of the series.

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BibTeX RIS

BibTeX

@misc{gong2026wielandt,
  title = {{Wielandt subgroups of nilpotent residuals and Fitting-height bounds}},
  author = {Lü Gong and Yan Yan and Enric Cosme Llópez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28082},
  eprint = {2610.28082},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28082},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Wielandt subgroups of nilpotent residuals and Fitting-height bounds
AU  - Lü Gong
AU  - Yan Yan
AU  - Enric Cosme Llópez
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28082
UR  - https://arxiv.org/abs/2610.28082
AB  - For a finite group G, let W(G) be the intersection of the normalizers of the nilpotent residuals of the subnormal subgroups of G, and let ℓ_W(G) be the length of the associated upper series. Thus W(G)=G exactly when all these residuals are normal in G. We prove that every finite soluble group satisfies ℓ_W(G)≤ h(G)≤4ℓ_W(G), where h(G) is the Fitting height, and that h(G)≤3ℓ_W(G) if |G| is odd or the Sylow $2$-subgroups of G are abelian. Both constants are attained at every length, the constant $4$ by {2,3}-groups. Moreover, every pair (n,h) with n≤ h≤4n occurs as (ℓ_W(G),h(G)). The constant $4$ comes from primitive groups. A soluble primitive group Vrtimes M satisfies W(Vrtimes M)=Vrtimes M if and only if every abelian subgroup of F(M) is cyclic. It then has Fitting height four if and only if F(M)=C× Q_8, with C cyclic, and M induces the full group Out(Q_8)≅ S_3. We also determine when the base group of a soluble regular wreath product is the first term of the series.
ER  -

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