Conjugacy Expansion in Alternating Groups: All Degrees up to 24 and Two Infinite Families of Classes

A finite group G is conjugacy-expansive if for every normal subset S and every conjugacy class C of G the normal set SC contains at least as many conjugacy classes of G as S does. The notion is due to Halasi, Maróti, Sidki and Bezerra, who asked whether every finite simple group is conjugacy-expansive (Kourovka Notebook, Problem 19.12); in Oberwolfach Report 27/2025, Dona asked for the case of the alternating groups. We give a partial answer: A_n is conjugacy-expansive for every 5 <= n <= 24 (exact computation with character-free certificates), and for every n >= 5 the defining condition holds for every class of support at most 6 and, for odd n, for both classes of n-cycles (the latter using the realizability theorem of Thom and Edmonds-Kulkarni-Stong); the general case n >= 25 stays open. The realizability theorem is quoted from Pervova and Petronio; the original papers of Thom and of Edmonds, Kulkarni and Stong were not read. By Hall's theorem, as observed by Halasi et al., the condition for a class C asks for a permutation π of the classes with π(K) ⊆ KC for every class K. For n <= 24 and every C we give explicit pairs x ∈ K, c ∈ C with xc ∈ π(K), which are checked by multiplying permutations; those for 19 <= n <= 24 were produced by an independent verification run and, because of their size, are regenerated by a program of the package. An exact character computation gives a second proof. For classes of small support the permutation is built from local configurations and a few explicit partners; for n-cycles it rests on a lemma saying that the two halves of a split class behave alike next to an n-cycle. We also give the short proofs of the known facts that S_n (n >= 3) and A_4 are not conjugacy-expansive. No printed answer to the question was found in the sources we searched; a search that finds nothing is not a proof of novelty, and no priority is claimed. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14299292-010 (Dona, Alternating groups as products of three conjugacy classes, Oberwolfach Report 27/2025).