On H-Bases and H-Decompositions of the Finite Solvable and Alternating Groups

✍ Scribed by G. Kozma; A. Lev

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 282 KB
• Volume: 49
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1994.1100

✦ Synopsis

Let (G) be a finite group such that every composition factor of (G) is either cyclic or isomorphic to the alternating group on (n) letters for some integer (n). Then for every positive integer (h) there is a subset (A \subseteq G) such that (|A| \leqslant(2 h-1)|G|^{1 / h}) and (A^{h}=G). The following generalization for the group (G) also holds: For every positive integer (h) and any nonnegative real numbers (\alpha_{1}, \alpha_{2}, \ldots, \alpha_{h}) so that (\alpha_{1}+\alpha_{2}+\cdots+) (\alpha_{h}=1) there are subsets (A_{1}, A_{2}, \ldots, A_{h} \subseteq G) such that (\left|A_{1}\right| \leqslant|G|^{x_{1}},\left|A_{i}\right| \leqslant 2|G|^{x_{i}}) for (2 \leqslant i \leqslant h) and (A_{1} A_{2} \cdots A_{h}=G). In particular, the above conclusions hold if (G) is a finite group and either (G) is an alternating group or (G) is solvable. (\quad 1994) Academic Press, Inc.