Some complete sets of complementary elements of the symmetric and the alternating group of the $n$-th degree

It is proved that some classes $\mathfrak{H}$ of conjugate elements in a symmetric and in an alternating group are complete sets of complementing elements, i.e., subsets such that for each non-identity element $A$ of the group there exists an element $B\in\mathfrak{H}$ such that $A$ and $B$ generate the group.