Is every derived subgroup of a finite $p$-group isomorphic to the Frattini subgroup of some finite $p$-group? (Kourovka 16.11)
Statement
Let $G$ be a finite $p$-group, with derived subgroup $[G,G]$, and let $\Phi(H)$ denote the Frattini subgroup of a finite $p$-group $H$. Does there always exist a finite $p$-group $H$ such that $\Phi(H)\cong[G,G]$? Equivalently: is every group that occurs as the derived subgroup of some finite $p$-group also isomorphic to the Frattini subgroup of some finite $p$-group?
Acceptance. FULLY RESOLVES: EITHER a proof that for every finite p-group G some finite p-group H satisfies Phi(H) isomorphic to [G,G] (ideally a uniform construction of H from G), OR a GAP-verifiable counterexample — an explicit finite p-group G whose derived subgroup [G,G] is provably not isomorphic to the Frattini subgroup of any finite p-group, with a machine-checkable certificate of non-realizability. ADVANCES: a certified small-order verification — for every p-group G up to order p^n (SmallGroups / p-group generation), an explicit H with Phi(H) isomorphic to [G,G], confirming 'yes' throughout that range and surfacing the derived subgroups D that are hardest to realize as Frattini subgroups (the natural counterexample candidates).
Background
Source: The Kourovka Notebook, No. 21, arXiv:1401.0300 (v44, June 2026), Problem 16.11, posed by Ya. G. Berkovich (16th Issue, 2006): 'Let G be a finite p-group. Does there always exist a finite p-group H such that Phi(H) is isomorphic to [G, G]?' Two classical strands feed the question but are studied separately: which finite p-groups occur as Frattini subgroups (Hobby's constraints — e.g. a Frattini subgroup cannot realize certain nonabelian obstructions), and which occur as derived subgroups (not all do; Blackburn characterized the 2-generated derived subgroups of p-groups). The combined realizability question — realize every derived subgroup as some group's Frattini subgroup — appears essentially untouched as a stated theorem. A natural attack is exhaustive search over the SmallGroups library / p-group generation: form D=[G,G] and hunt for H with Phi(H) isomorphic to D; a single non-realizable D is a finite counterexample certificate. Vetted open as of 2026-07-06: not asterisked or commented in v44 and no dedicated resolution found — but the literature is thin on both sides, so open-status confidence is only MEDIUM (a resolution could follow 'for free' from a broader Frattini-realizability result, and this should be re-checked before heavy investment). The statement follows the notebook verbatim; looser 'every finite group as derived-subgroup-equals-Frattini' paraphrases are inaccurate.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | The Kourovka Notebook No. 21 (v44, June 2026) — Problem 16.11 (Ya. G. Berkovich) | arxiv |
| REF-02 | The Kourovka Notebook — official site | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.