User ahmed elsawy - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:43:47Z http://mathoverflow.net/feeds/user/21286 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60750/profinite-completion-of-a-semidirect-product/88091#88091 Answer by Ahmed Elsawy for Profinite completion of a semidirect product Ahmed Elsawy 2012-02-10T11:08:05Z 2012-02-10T11:08:05Z <p>Let $\mathscr{P}$ be any property such that whenever a group has $\mathscr{P}$ then all its subgroups also have $\mathscr{P}$. In [1] Theorem 3.1, K. W. Gruenberg has proved that if the wreath product $W= A \wr B$, is residually $\mathscr{P}$, then either $B$ is $\mathscr{P}$ or $A$ is abelian. </p> <p>Consider $W= S_3 \wr \mathbb{Z}$, where $S_3$ is the symmetric group of degree 3. Since $S_3$ is not abelian, $\mathbb{Z}$ is not finite, and the subgroup of any finite group is finite, the group $W$ is not RF.</p> <p>Clearly, $W= \prod_{i \in \mathbb{Z}} S_3 \rtimes \mathbb{Z}$, where $\mathbb{Z}$ and $\prod_{i \in \mathbb{Z}} S_3$ are residually finite.</p> <p>[1] K. W. Gruenberg, Residual properties of infinite soluble groups}, Prec. London Math. Soc., Ser. 3, 7 (1957), 29--62.</p>