If G is a group and T is a
nonempty subset of G, we say that T divides G if and only if G contains a subset S
such that every element of G has a unique representation as ts with t in T, s in S, in
which case we write T ⋅ S = G. We study the case where G is Σ_{n}, the symmetric
group on n symbols and T is the set consisting of the identity and all transpositions
in Σ_{n}.
The problem may be given a combinatorial setting as follows: For x, y in
Σ_{n}, let d(x,y) be the minimum number of transpositions needed to write
xy^{−1}. One verifies that d converts Σ_{n} into a metric space, and that T divides
Σ_{n} if and only if Σ_{n} can be covered by disjoint closed spheres of radius
one.
We use the irreducible characters of Σ_{n}, together with judiciously selected
permutation representations of Σ_{n}, to prove the following result.
Theorem. If 1 + (n(n − 1))∕2 is divisible by a prime exceeding + 2, then T
does not divide Σ_{n}.
