Kourovka Notebook Problem 21.142

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.

Write $a^g=g^{-1}ag$. A group $G$ is invariably generated by $a,b$ if $\langle a^g,b^h\rangle=G$ for every $g,h\in G$. An embedding is an injective group homomorphism.

Statement recorded in the supplied source.

For each pair of distinct primes $p,q$ and every finite group $F$, does there exist a finite group $G$ containing an embedded copy of $F$ and elements $a,b\in G$ of orders $p,q$, respectively, that invariably generate $G$? [S1][S2]

Short English statement

Can every finite group be embedded into a finite group generated by any independent conjugates of two elements having prescribed distinct prime orders?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2651 (KOU-21.142). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 E. I. Khukhro and V. D. Mazurov (editors), The Kourovka Notebook, Unsolved Problems in Group Theory, No.21, version48 (6 October2026), Problem 21.142 and its remarks. https://arxiv.org/abs/1401.0300v48.