Conjugacy Classes and Characters of Finite p-Groups

Definitions and mathematical statement

Let $p$ be a prime and $P$ a finite $p$-group. The conjugacy class of $x\in P$ is $x^P=\{g xg^{-1}:g\in P\}$, and the class number is \[ h_P=\#\{x^P:x\in P\}. \] In the customary ordinary-character interpretation, $\operatorname{Irr}(P)$ is the set of irreducible complex characters of $P$; $\chi(1)$ is the degree of $\chi$. For every integer $k\geq0$, put \[ a_k(P)=\#\{\chi\in\operatorname{Irr}(P):\chi(1)=p^k\}. \] The four requested tasks are to develop general methods that parametrize the $P$-conjugacy classes, compute $h_P$, describe $\operatorname{Irr}(P)$, and compute each $a_k(P)$.

This is a structural and computational research program, not one stated numerical formula or a claim that finite-group data cannot be obtained by finite enumeration. The source supplies no input presentation, complexity target, preferred parametrizing objects, or distinguished family of $p$-groups. Its literature assessment refers to a generalized Cayley approach but gives neither the construction nor the classes to which it applies. These details are needed to formulate a specific uniform method or theorem.

Short English statement

Develop general structural methods for conjugacy classes, their number, irreducible characters, and the number of characters of each degree $p^k$ in finite $p$-groups.

Sources

• Local UnsolvedMath registry, permanent ID $30000985$, source code catalogue entry, "Conjugacy Classes and Characters of Finite p-Groups." No individual proposer is supplied.
• Profinite and Asymptotic Group Theory (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/28, pp. 1537--1588: https://doi.org/10.4171/owr/2008/28. This is the sole supplied external reference.
• Supplied literature triage, checked 2026-08-21: status "open." It calls this a list of general character-theory questions rather than a single assertion, and reports that the classes admitting the generalized Cayley approach are not known in the report. This dated assessment derives from supplied OWR curation; time-sensitive status and cited claims require independent verification.