Olson's Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be any finite group, written multiplicatively, with identity $1_G$ and $|G|=n\ge1$. A sequence may repeat elements. A subsequence retains the order of its indices. Also define $z(G)$ as the least $m$ such that every length-$m$ sequence in $G$ has an ordered subsequence of length $n$ with product $1_G$.

Statement recorded in the supplied source.

For every sequence $a_1,\ldots,a_{2n-1}$ in $G$, must there exist \[ 1\le j_1<j_2<\cdots<j_n\le2n-1, \qquad a_{j_1}a_{j_2}\cdots a_{j_n}=1_G? \] Equivalently, is $z(G)\le2|G|-1$ for every finite group? The source also asks for improved general bounds on $z(G)$ beyond $z(G)\le|G|(|G|-1)+1$. Multiplication follows the original order even for nonabelian groups. [S2]

Short English statement

Find a product-one subsequence of group-order length without rearranging its terms, and improve bounds on the minimum sequence length guaranteeing one.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3376 (OPG-155). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Olson's Conjecture, OPG-155. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/olsons_conjecture.