Automorphisms of the symmetric group

An automorphism preserving the transposition class is inner: commuting relations recover the vertex stars of the complete graph, with a small-case adjustment in degrees three and four. An involution class has the same size as the transposition class only in degree six, when triple transpositions also have size fifteen. Hence outer automorphism groups of symmetric groups are trivial outside degree six and have size at most two in degree six.

// Target: combinatorics.bigb

A permutation of order two is a product of $k$ disjoint transpositions. Its conjugacy class has the displayed size. For $k\ge 2$, equality with the transposition class requires $(n-2)!/(n-2k)!=2^{k-1}k!$. The left side grows strictly with $n$. At $n=2k$, its ratio to the right side is one only for $k=3$; at $k=2$ it jumps from below to above one between $n=4$ and $n=5$, and for $k\ge 4$ it is already greater than one. Thus degree six is the only possible exceptional degree for an group automorphism not preserving transpositions.