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 \(5\le n\le 24\) (exact computation with character-free certificates), and for every \(n\ge5\) 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 \(n\ge25\) 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 \(\pi\) of the classes with \(\pi(K)\subseteq KC\) for every class \(K\). For \(n\le 24\) and every \(C\) we give explicit pairs \(x\in K\), \(c\in C\) with \(xc\in\pi(K)\), which are checked by multiplying permutations; those for \(19\le n\le 24\) 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 \(S_n\) \((n\ge3)\) 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.