Let S n be the symmetric group on the set X = {1, 2, ....., n}. A subset S of S n is intersecting if for any two permutations g and h in S, g(x) = h(x) for some x ∈ X (that is g and h agree on x). M. Deza and P. Frankl [4] proved that if S ⊆ S n is intersecting then |S| ≤ (n − 1)!. This bound is met by taking S to be a coset of a stabilizer of a point. We show that these are the only largest intersecting sets of permutations.

We analyse a probabilistic argument that gives a semi-random construction for a permutation code on n symbols with distance n − s and size (s!(log n) 1/2 ), and a bound on the covering radius for sets of permutations in terms of a certain frequency parameter.

Let $S_{n}$ denote the symmetric group on $[n]=\{1, \ldots, n\}$. A family $I \subseteq S_{n}$ is intersecting if any two elements of $I$ have at least one common entry. It is known that the only intersecting families of maximal size in $S_{n}$ are the cosets of point stabilizers. We show that, under mild restrictions, analogous results hold for the alternating group and the direct product of symmetric groups.

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Let Sym([n]) denote the collection of all permutations of) is a family of permutations such that any two of its elements (when written in its cycle decomposition) have at least t cycles in common. We prove that for sufficiently large n, |A| (n − t)! with equality if and only if A is the stabilizer of t fixed points. Similarly, let B(n) denote the collection of all set partitions of [n] and suppose A ⊆ B(n) is a family of set partitions such that any two of its elements have at least t blocks in common. It is proved that, for sufficiently large n, |A| B n−t with equality if and only if A consists of all set partitions with t fixed singletons, where B n is the nth Bell number.