Thin systems of generators of groups
Algebra and discrete mathematics, Tome 9 (2010) no. 2, pp. 108-114
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A subset $T$ of a group $G$ with the identity $e$ is called $k$-thin $(k\in{\mathbb N})$ if $|A\cap gA|\leqslant k$, $|A\cap Ag|\leqslant k$ for every $g\in G$, $g\ne e$. We show that every infinite group $G$ can be generated by some 2-thin subset. Moreover, if $G$ is either Abelian or a torsion group without elements of order 2, then there exists a 1-thin system of generators of $G$. For every infinite group $G$, there exist a 2-thin subset $X$ such that $G=XX^{-1}\cup X^{-1}X$, and a 4-thin subset $Y$ such that $G=YY^{-1}$.
Keywords:
small, $P$-small, $k$-thin subsets of groups.
@article{ADM_2010_9_2_a7,
author = {Ievgen Lutsenko},
title = {Thin systems of generators of groups},
journal = {Algebra and discrete mathematics},
pages = {108--114},
publisher = {mathdoc},
volume = {9},
number = {2},
year = {2010},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ADM_2010_9_2_a7/}
}
Ievgen Lutsenko. Thin systems of generators of groups. Algebra and discrete mathematics, Tome 9 (2010) no. 2, pp. 108-114. http://geodesic.mathdoc.fr/item/ADM_2010_9_2_a7/