For distinct primes p and q, the alternating group A_n, for every sufficiently large n, does not embed into any finite group invariably generated by an order-p and an order-q element.
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Alternating Groups and Embeddings into Groups Invariably Generated by Two Prime-Order Elements
For distinct primes p and q, the alternating group A_n, for every sufficiently large n, does not embed into any finite group invariably generated by an order-p and an order-q element.