a transitive subgroup of Sn, for some n. Cayley's theorem Cayley's theorem states that every group G is isomorphic to a subgroup of some symmetric group. In particular, one may take a subgroup of the symmetric group on the elements of G, since every group acts on itself faithfully by (left or right) multiplication. Cyclic subgroups Cyclic groups are those that are generated by a single permutation. When a permutation is represented in cycle notation, the order of the cyclic subgroup that it generates is the least common multiple of the lengths of its cycles. For example, in S, one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S are generated by elements like (123)(45) that have one cycle of length 3 and another cycle of length 2. This rules out many groups as possible subgroups of symmetric groups of a given size. For example, S has no subgroup of order 15 (a divisor of the order of S), because the only group of order 15 is the cyclic group. The largest possible order of a cyclic subgroup (equivalently, the largest possible order of an element in S) is given by Landau's function. Automorphism group For , Sn is a complete group: its center and outer automorphism group are both trivial. For , the automorphism group is trivial, but S2 is not trivial: it is isomorphic to C2, which is abelian, and hence the center is the whole group. For , it has an outer automorphism of order 2: , and the automorphism group is a semidirect product . In fact, for any set X of cardinality other than 6, every automorphism of the symmetric group on X is inner, a result first due to according to . Homology The group homology of Sn is quite regular and stabilizes: the first homology (concretely, the abelianization) is: The first homology group is the abelianization, and corresponds to the sign map Sn → S2 which is the abelianization for n ≥ 2; for n < 2 the symmetric group is trivial. This homology is easily computed as follows: Sn is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps are to S2 and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps send an involution