Kourovka Problem 21.142 — invariable generation embedding

Solved with Albilich (AI). Human authors: Ting Gong, Yong Yang, Michael Ruofan Zeng. AI system: Albilich.

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Answered negatively. For every pair of distinct primes $p \neq q$, there is a finite group that embeds into no finite group invariably generated by an element of order $p$ and an element of order $q$.

Theorem. Let $p \neq q$ be fixed primes. Then there exists an alternating group which does not embed into any finite group invariably generated by an element of order $p$ and an element of order $q$.

The counterexample is an alternating group $A_{n_0}$, with $n_0$ chosen sufficiently large in terms of $p$ and $q$.

Proof sketch

Suppose $H$ is a finite group invariably generated by elements of orders $p$ and $q$, and let $S$ be a nonabelian composition factor of $H$. Some chief factor of $H$ has the form $S^t$, and conjugation on it gives a quotient $G$ with

$$S^t \le G \le \operatorname{Aut}(S) \wr \operatorname{Sym}(t).$$

Invariable generation passes to quotients, and since $G \ge S^t$ is noncyclic neither generator image is trivial, so their orders are exactly $p$ and $q$. The problem therefore reduces to a uniform bound $B(p,q)$, depending only on $p$ and $q$, on the degree of any alternating section $A_n$ of $S$.

That bound is obtained in two steps. A wreath-product conjugacy lemma, applied through stabilisers of subsets of size $pq$, shows that a fixed proper intransitive subgroup meets every conjugacy class of elements of order $p$ or $q$; this excludes alternating groups of large degree, and classical groups of large Lie rank are excluded separately. For the remaining simple groups the classification of finite simple groups bounds the degree of a faithful projective representation of $S$, and Theorem A of Collins then bounds $n$ whenever $A_n$ is a section of $S$. Taking $n_0 > \max\{B(p,q), 4\}$, the group $A_{n_0}$ embeds into no such $H$, since otherwise it would be a section of a composition factor of $H$, contradicting the definition of $B(p,q)$.

Reference

Ting Gong, Yong Yang, Michael Ruofan Zeng, Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements, arXiv:2608.00703 (submitted 1 August 2026, 14 pp.). Theorem 1.1.

Attribution

Human authors: Ting Gong (University of Washington), Yong Yang (Texas State University), Michael Ruofan Zeng (University of Washington).

AI system: Albilich — an open-source agentic proof-state harness developed by the authors, which produced the counterexample route and a human-readable PDF. The paper was then written up by the human authors, and all statements and proofs were then checked and revised by them. Albilich is described in Ting Gong, Michael Ruofan Zeng, Yong Yang, Albilich: Steerable Proof-State Orchestration for LLM-Based Mathematical Research with CAS Integration, arXiv:2607.27705; the run for this problem used 80 child sessions harnessing GPT-5.6 Sol and produced one integrated proof route to the root statement.