On a subgroup contained in some words with a bounded length
The author proves the following result: Let \(G\) be a group of order \(\geq 2\) and \(A\), \(B\) two finite nonvoid subsets of \(G\) such that \(1\not\in B\). Then either \(| A\cup B\cup AB|\geq| A|+| B|\) or there exists a nonnull subgroup contained in \(A\cup B\cup AB\). As an application of the above result, the author also obtains the following proposition: Let \(A_ 1,A_ 2,\dots, A_ k\) be subsets of a finite group \(G\) such that \(1\not\in A_ i\), \(2\leq i\leq k\), and \(| A_ 1|+| A_ 2|+\dots+ | A_ k|\geq| G|\). The union of sets of the form \(A_{i_ 1},\dots, A_{i_ j}\) for \(1\leq i_ 1<i_ 2<\dots <i_ j\leq k\) must include a nonnull subgroup. In particular if \(B\) is a subset of \(G\setminus 1\) such that \(k| B|\geq | G|\), the set \(B\cup B^ 2\cup \dots\cup B^ k\) must contain a nonnull subgroup. The above proposition generalizes a theorem of \textit{G. T. Diderrich}, who obtained the same conclusion in the case \(| A_ 1|=| A_ 2|= \dots=| A_ k|=1\) [Proc. Am. Math. Soc. 38, 443-451 (1973; Zbl 0266.20041)].
scientific article; zbMATH DE number 89923