Near-factors of Finite Groups

Let \(S\) and \(T\) be subsets of a finite group \(G with identity \(e\), We write \(G-e =ST\) if every non-identity element \(g\) can be written uniquely as \(g = st\) with \(s \in S\) and \(t \in T\). These near-factorizations are motivated by the combinatorial problem of
finding \((0 , 1)\)-matrix factorizations of the matrix \(J-I\). We derive some results on near-factors \(S\) and \(T\). For example, \(S\) and \(T\) each generate \(G\). Also, if \(G\) is abelian, then the automorphism \(g \rightarrow g^{-1}\) is a multiplier of both \(S\) and \(T\). If the elementary abelian group \(C_p^n\) (\(p\) an odd prime) is a homomorphic image of \(G\), then \(|S|^{p-1} \equiv |T|^{p-1} \equiv 1
(mod p)\). These structure theorems suggest that noncyclic abelian groups rarely have near-factorizations. Constructions of near-factorizations are given for cyclic groups and dihedral groups.