Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements

Abstract

For any distinct primes $p$ and $q$, we prove that there is a finite group which does not embed into any finite group invariably generated by an element of order $p$ and an element of order $q$. This gives a negative answer to Problem 21.142 of the Kourovka Notebook \cite{kourovka21}. In fact, for every fixed pair $p,q$, the group $A_n$ cannot embed into such a group once $n$ is sufficiently large in terms of $p$ and $q$.

Disclosure

“e write Sn and An for the symmetric and alter- nating groups on n letters, and Sym(t) for the symmetric group acting on t direct factors. Statement on the use of AI The authors used Albilich [GZY26], a generative AI system developed by the authors, during the development of the argument. All statements and proofs were checked and revised by the authors, who take full responsibility for the contents of the paper. 2. Definit”

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Classification

Proof ideas or individual proof-step assistance

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8

Verified

2026-08-23T19:16:56.533825Z

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