Direct products of subsets in a finite Abelian group. (Q2434157)

For a finite Abelian group \(G\) and subsets \(A_1,\ldots,A_n\subseteq G\) we say that \(A_1A_2\cdots A_n\) is a direct product (or factorization) if every element of \(A_1\cdots A_n\) can be uniquely written in the form \(a_1a_2\cdots a_n\) for some \(a_1\in A_1,\ldots,a_n\in A_n\). A subset \(A\) of \(G\) is called cyclic if \(A=\{e,a,a^2,\ldots,a^{r-1}\}\) for some \(a\in G\) and \(r\leq |a|\). A fundamental theorem of \textit{G. Hajós} [Math. Z. 47, 427-467 (1941; Zbl 0025.25401)] asserts that if \(G=A_1A_2\ldots A_n\) is a factorization of \(G\) into cyclic subsets, then at least one of the factors \(A_1,\ldots, A_n\) must be a (cyclic) subgroup of \(G\). The paper under review deals with some extensions of Hajós' theorem. First, the authors introduce some terminology. Let \(A\subseteq G\) be a subset of \(G\). Then \(A\) is called a distorted cyclic subset if \(A=\{e,a,\ldots,a^{t-1},a^td,a^{t+1},\ldots,a^{r-1}\}\) for some \(a,d\in G\), \(t\leq r-1<|a|\) such that \(a^td\neq a^j\) for each \(0\leq j\leq r-1\). \(A\) is simulated, if there is a subgroup \(H\leq G\) such that \(|A|=|H|\) and \(|A\cap H|+1\geq |A|\). \(A\) is periodic, if its stabilizer \(\{g\in G\mid gA=A\}\neq e\). Finally, a factorization \(G=A_1\cdots A_n\) is normalized if \(e\in A_i\) for each \(1\leq i\leq n\). Now, the following generalization of Hajós' theorem is proved. Let \(G=B_1\cdots B_nD\) be a normalized factorization of \(G\), where \(B_1,\ldots,B_n\) are distorted cyclic subsets and \(B_1\cdots B_n\) is periodic. Then at least one of the factors \(B_1,\ldots,B_n\) is periodic. The main step in the proof is the following lemma confirmed by the authors using character theory. Let \(B\subseteq G\) be a distorted cyclic subset and \(A\subseteq G\) be the cyclic subset associated with \(B\). Assuming that \(|A|\geq 4\), if \(G=BC\) is a normalized factorization, then \(G=AC\) is also a normalized factorization. In the second part of the proof the authors give some sufficient conditions when a direct product of some cyclic or simulated subsets of \(G\) is a subgroup of \(G\).