A finite $p$-group of odd order with $|\mathrm{Aut}\,G|=|G|$: does one exist? (Kourovka 16.63, MacHale)
Statement
For a finite group $G$ let $\mathrm{Aut}\,G$ be its automorphism group. Does there exist a non-trivial finite $p$-group $G$ of ODD order with $|\mathrm{Aut}\,G|=|G|$? Exhibit such a group, or prove that every non-trivial odd-order finite $p$-group has $|\mathrm{Aut}\,G|\ne|G|$ (equivalently $|\mathrm{Aut}\,G|>|G|$).
Acceptance. FULLY RESOLVES: either (a) an explicit odd-order $p$-group $G$ (power-commutator presentation) with a machine-verified computation showing $|\mathrm{Aut}\,G|=|G|$ (a finite reproducible certificate: GAP AutomorphismGroup + order check), or (b) a proof that $|\mathrm{Aut}\,G|>|G|$ for all non-trivial odd-order $p$-groups. ADVANCES: extend the verified range (prove $|\mathrm{Aut}\,G|>|G|$ for all odd-order $p$-groups up to a new order bound, with the enumeration certificate), or settle it within a structurally defined subclass (e.g. all groups of maximal class, or of a fixed coclass). Provide the group data and the automorphism-order script.
Background
Problem 16.63 of the Kourovka Notebook (D. MacHale, 16th issue 2006; cf. Archive 12.77). For $p=2$ such groups exist — e.g. the dihedral group $D_8$ of order 8 has $|\mathrm{Aut}\,D_8|=8$ — so only the odd case is in question. Best-known frontier: numerous partial results force $|\mathrm{Aut}\,G|>|G|$ under structural hypotheses (class 2, $p$-central, groups with a nontrivial abelian direct factor); Helleloid-Martin (J. Algebra 312 (2007), 'The automorphism group of a finite p-group is almost always a p-group') show $\mathrm{Aut}\,G$ is a $p$-group for most $p$-groups but do not settle equality; the abelian classification of $A$ with $|\mathrm{Aut}\,A|=|A|$ is known, and the inversion-free / N.I.-group line (Curran, Heffernan & MacHale) gives $3^7$ as the smallest order of a nontrivial odd-order group realized as a full automorphism group. No odd $p$-group with equality is known, nor a proof of impossibility. An attacker needs: (i) GAP/Magma over the SmallGroups and $p$-group-generation libraries for $p=3,5,7$ and orders up to $\sim p^{9\text{-}10}$, computing $|\mathrm{Aut}\,G|$ directly; (ii) for the negative side, structural lower bounds on $|\mathrm{Aut}\,G|/|G|$ from the $\mathrm{GL}$-action on $G/\Phi(G)$ and the automizer of $Z(G)$, which are large for odd $p$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Unsolved Problems in Group Theory: The Kourovka Notebook (No. 21) | arxiv |
| REF-02 | The automorphism group of a finite p-group is almost always a p-group (Helleloid-Martin, 2007) | paper |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.