Near-factors of Finite Groups

Abstract

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 $g\to g^{-1}$ 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 $|S{|}^{p-1}\equiv |T{|}^{p-1}\equiv 1(modp)$. These structure theorems suggest that noncyclic abelian groups rarely have near-factorizations. Constructions of near-factorizations are given for cyclic groups and dihedral groups.