= Automorphisms of alternating groups from triple supports
{title2=$\operatorname{Aut}(A_n)\cong S_n\quad(n\ge5,\ n\ne6)$}
Except in degree six, the $3$-cycles are characterized among order-three elements by the largest <centralizer>. Their cyclic subgroups correspond to three-element supports. Two different such subgroups commute exactly for disjoint supports; with one common point their generator products have order five, and with two common points those products have order two or three. Thus an automorphism induces an automorphism of $J(n,3)$. The <maximal cliques of a Johnson graph> reconstruct a permutation of the points. After undoing that permutation, an automorphism fixes every cyclic $3$-subgroup. Product orders force its choices of generator inversion to be consistent on the connected graph, and inversion of every $3$-cycle would reverse a noncommuting product. Hence it fixes all $3$-cycles, which generate $A_n$.
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