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For a positive integer $n$, let $p_n$ denote the $n$-th prime number. Further let $f: {\rm Sym}(\mathbb{N}) \rightarrow {\rm Sym}(\mathbb{N})$ be the monomorphism which maps a permutation $\sigma$ to …

The smallest $n$ for which there exist sequences as asked for is $n = 7$: $(1,2,3,4,5,6,7) \cdot (1,2) \cdot (1,7,6,5,4,3,2) \cdot (1,2) \cdot (1,2,3,4,5,6,7) \cdot (1,2) \cdot$ $(1,7,6,5,4,3, …

If you are also interested in problems of that type where $n = \infty$: Given a mapping $f: \mathbb{N} \rightarrow \mathbb{N}$ from the natural numbers to themselves, it is often a notoriously hard pr …

For the sake of simplicity, consider only the case $d=2$. In this case, two pairs $(a,b), (a,c) \in {\rm S}_n^2$ lie in the same orbit if and only if there is a permutation $\pi$ in the centralizer of …

The GAP convention is to multiply permutations from the left to the right, i.e. $(1,2) \cdot (1,3) = (1,2,3)$, to write down each cycle with the smallest moved point first and to sort cycles in ascending …