The Rédei property

Abstract

By Rédei’s theorem, if a finite abelian group is factored into normalized subsets of prime cardinality, then at least one of the factors is a subgroup of the group. The special case when G is of type (p, p) plays an important part of the proof of the general case and has interesting geometric and combinatorial applications. In connection with this result, Rédei formulated the next conjecture which also appears as Problem 5 in the Open Problems section of his book:

Let p be a prime and G be a group of type (p, p, p). Consider a normalized factorization G = AB of G, where |A| = p and |B| = p². Then either 〈A〉 ≠ G or 〈B〉 ≠ G.