A Necessary Condition for Transitivity of a Finite Permutation Group

Suppose the group G is generated by permutations gvg2,...,g, acting on a set CI of size n, such that 8182--8, ' s the identity permutation. If the generator gt has exactly ci cycles (for 1 ^ / < s), and G is transitive on Ci, then n(s—2)—YJ'(-I c, + 2 is a non-negative even integer. This is proved using an elementary graph-theoretic argument. In this paper we give a simple proof of a necessary condition involving the number of cycles of any generating set for a finite permutation group. Specifically, we prove the following: THEOREM. Suppose G is a group of permutations of a set CI of size n, and G is generated by the elements g1,g2, ...,gs, whose product is the identity permutation. If the generator gt has exactly c{ cycles on CI {for 1 ̂ / ^ s) and G is transitive on CI, then n(s—2) — X!i-i( + 2 must be a non-negative integer. Naturally we assume s ^ 2. The hypothesis g1g2--g8 = 1 simply requires that gg is the inverse of the product of the first s— 1 generators (which together generate G themselves). By 'cycles' we mean 1-cycles (fixed points) as well as 2-cycles (transpositions), 3-cycles, and so on. It is an easy exercise to show that if the permutation g o n f l has c cycles, then g is even if and only if c = n modulo 2. Further, because the product of the generators 8i>82>--'>8* °f G i the identity, all but an even number of them will be even permutations. It therefore follows that Y£-\ < = ns odulo 2, and hence