Does such a group exist? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T05:54:56Z http://mathoverflow.net/feeds/question/99695 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99695/does-such-a-group-exist Does such a group exist? elad 2012-06-15T11:13:33Z 2012-06-15T16:29:41Z <p>A group with infinite normal subgroups: $G\varsupsetneqq G_{E}\varsupsetneqq G_{1}\varsupsetneqq G_{2}\varsupsetneqq....$ such that there exist a subset $B\subset G$ satisfying: 1) $\forall x,y\in B\,\, x\neq y\Rightarrow x\cdot y^{-1}\notin G_{E}$</p> <p>2) $\forall n\in\mathbb{N}$ $G/G_{n}=\{{\overline{x}\cdot\overline{y}\cdot\overline{z}\,|\, x,y,z\in B\}}$</p> http://mathoverflow.net/questions/99695/does-such-a-group-exist/99716#99716 Answer by Will Sawin for Does such a group exist? Will Sawin 2012-06-15T15:45:45Z 2012-06-15T16:29:41Z <p>$\mathbb Z \times \mathbb Z$ will do. Take $G_E$ to consist of the elements whose first coordinate is $0$ and $G_n$ to be the subgroup of multiples of $2^n$ inside $G_E$. What $G_n$ are is immaterial, since we will make $G$ itself be covered by <code>$\{x\cdot y\cdot z| x,y,z\in B\}$</code>.</p> <p>We will do this by building up $B$ step-by-step. At any given time, it contains finitely many elements. Choose some element of $G$ that is not in <code>$\{x\cdot y\cdot z| x,y,z\in B\}$</code>. We will add $3$ elements to $B$ so that it is. We can do this while preserving the property (1), since that just means that the first coordinates of all the elements are different. So choose two very large positive first coordinates, and then a third very large negative first coordinate so that they add up to the desired value. If they are sufficiently large they will not intersect past first coordinates. Then choose second coordinates in any way that adds up to the desired value.</p> <p>Repeat this process until every element of $G$ is covered. Then clearly $G/G_n$ is covered as well.</p> <p>This should work any time $G/G_E$ is infinite (of at least the cardinality of $G_E$).</p>