Some Questions — Question 51

Definitions and mathematical statement

Let $F_r$ be the free group on variables $x_1,\ldots,x_r$, where $r\geq1$ is finite. A word $w\in F_r\setminus\{1\}$ is a law in a finite group $P$ if $w(g_1,\ldots,g_r)=1$ for every tuple $(g_1,\ldots,g_r)\in P^r$. Its length is the number of letters in its freely reduced form, counting a variable or its inverse as one letter.

For each integer $n\geq1$, define \[ L_2(n)=\min\{|w|:w\ne1\text{ is a law in every group }P \text{ with }|P|=2^n\}. \] The number of variables may vary in this minimization, but one word must work for all the groups of that order and every assignment in each group. Determine the asymptotic growth of $L_2(n)$ as $n\to\infty$, ideally with matching upper and lower bounds.

The power word $x^{2^n}$ shows that the minimum is finite. A law in all groups of order exactly $2^n$ is also a law in every smaller $2$-group: embed a group of order $2^m$ in its direct product with an elementary abelian group of order $2^{n-m}$. Thus replacing exactly by at most does not change the extremal function. This problem differs from finding the shortest law in one iterated wreath product, and word length is measured in the variables rather than in a chosen generating set of the tested group.

Short English statement

What is the shortest word that is an identity in every group of order two to the n?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1200051 (AMR-011-0051), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Abert - Some questions (pdf) (2010), Question 51. Original source indexed by AMR-011-0051. https://www.renyi.hu/~abert/questions.pdf.