A finite $p$-group with nontrivial Hughes subgroup of index exactly $p^3$ (Kourovka 8.85, Khukhro)
Statement
For a finite $p$-group $G$, its Hughes subgroup is $H_p(G)=\langle x\in G : x^p\ne 1\rangle$, generated by all elements of order $\ne p$. Construct a finite $p$-group $G$ for which $H_p(G)$ is nontrivial and $[G:H_p(G)]=p^3$, or prove that no finite $p$-group has $1\ne H_p(G)$ with $[G:H_p(G)]=p^3$.
Acceptance. FULLY RESOLVES: either (a) an explicit finite $p$-group $G$ (power-commutator presentation) with a machine-verified certificate that $H_p(G)\ne 1$ and $[G:H_p(G)]=p^3$ (compute the subgroup generated by all elements of order $\ne p$ and its index in GAP), or (b) a proof that this index cannot equal $p^3$ for any finite $p$-group with $H_p(G)\ne 1$. ADVANCES: rule out index $p^3$ for a structurally defined subclass (e.g. metabelian, or fixed small class/exponent), or push the known index-$p^2$ families toward the $p^3$ boundary, each with a reproducible GAP certificate. Provide the group and the verification script.
Background
Problem 8.85 of the Kourovka Notebook (E. I. Khukhro, 8th issue 1982). Hughes' 1957 conjecture — that $[G:H_p(G)]\in\{1,p\}$ always — is FALSE: G. E. Wall exhibited a 3-generator finite 5-group with $[G:H_p(G)]=5^2$; Hughes himself proved the conjecture for $p=2$ and Straus-Szekeres for $p=3$. Index-$p^2$ counterexamples are known for primes $5\le p\le 19$, and in EVERY known counterexample $[G:H_p(G)]=p^2$ with $\exp(G)=p^2$. Whether index $p^3$ can occur is the open frontier (it is expected but unrealized). Recent adjacent work: 'On the Hughes conjecture for some finite $p$-groups' (Ann. Mat. Pura Appl. 202 (2023)); the tidy / absolutely-regular $p$-group classifications (Bull. Aust. Math. Soc.); generalizations of the Hughes subgroup (arXiv:1809.07564). An attacker needs: (i) the Lie-ring / power-commutator machinery behind the known index-$p^2$ 'anti-Hughes' groups, deformed to force one further layer of order-$\ne p$ generation; (ii) GAP's $p$-group functions to compute $H_p(G)$ and its index on candidate groups (a finite check). Nonexistence would be a bounded-class structural theorem.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Unsolved Problems in Group Theory: The Kourovka Notebook (No. 21) | arxiv |
| REF-02 | On the Hughes conjecture for some finite p-groups (2023) | paper |
| REF-03 | A Generalization of the Hughes Subgroup | arxiv |
Investigations · 0
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