= Transposition changes the cycle count by one
{title2=$\ell(\tau\sigma)=\ell(\sigma)\pm1$}
Left multiplication of a <permutation> by a <transposition> $(a\ b)$ swaps the two target labels $a,b$. If they occur in different cycles, $(a\ x_1\ldots x_p)(b\ y_1\ldots y_q)$ becomes the single cycle $(a\ x_1\ldots x_p\ b\ y_1\ldots y_q)$. If they occur in the same cycle, that formula reverses and splits the cycle in two. All other cycles are unchanged, so the total number of cycles, including fixed points, decreases or increases by exactly one. Empty intermediate lists yield one-cycles and require no exceptional case.
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