Kourovka Notebook Problem 3.60

Definitions and mathematical statement

Let $G$ be a finite group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$: if $|G|=p^a m$, where $a\geq 0$ and $p\nmid m$, then $|P|=p^a$.

Write $\ell_p(G)$ provisionally for the notion of $p$-length of an arbitrary finite group attributed in the source to L. A. Shemetkov, Math. USSR Sbornik, 1 (1968), 83--92. Write $c_p(P),d_p(P),e_p(P)$ for the three invariants of the Sylow subgroup mentioned in the source. These symbols are placeholders: the supplied record gives neither the definition of this generalized $p$-length nor the definitions of these invariants.

Research problem (L. A. Shemetkov, 1969). Investigate the relations between $\ell_p(G)$ and $c_p(P),d_p(P),e_p(P)$ for arbitrary finite groups $G$.

The source poses an investigation, without specifying an inequality, equivalence, or other proposition to establish. It does not restrict $G$ to be $p$-soluble, so a definition of $p$-length confined to that class would not specify the stated problem. The intended definitions and the conventions when $P$ is trivial must be recovered before a fully precise formulation can be given; the source does not explicitly restrict to primes dividing $|G|$.

Short English statement

Shemetkov asks how his notion of $p$-length for an arbitrary finite group relates to three invariants $c_p,d_p,e_p$ of its Sylow $p$-subgroup. The supplied record does not define these quantities or specify a particular relation to prove, so this remains an underdetermined research question.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700208$, exact source problem code catalogue entry, title Kourovka Notebook Problem 3.60; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is supplied status metadata, not an independently verified mathematical assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 3.60, attributed to L. A. Shemetkov, 3rd issue (1969). The supplied record identifies the 21st issue edition (2026), arXiv:1401.0300v48; its collection description names E. I. Khukhro and V. D. Mazurov as editors.
• L. A. Shemetkov, Math. USSR Sbornik, 1 (1968), 83--92, cited by the supplied statement for the definition of $p$-length of an arbitrary finite group. The record supplies no article title or definition. Bibliographic details are reproduced as supplied and have not been independently verified.