dear all, Let $H=Z_2 \wr Z_2 \wr...\wr Z_2$ ( r times), I need to know the structure of H as a matrix group. Thanks in advance
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$\begingroup$ You should specify which field(s) you want to work over. $\endgroup$– Geoff RobinsonCommented Apr 13, 2013 at 9:30
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1$\begingroup$ I am not convinced that $H$ is uniquely defined. The wreath product is an associative operation on permutation groups, but not on abstract groups, and $Z_2$ looks like an abstract group rather than a permutatino group. $\endgroup$– Derek HoltCommented Apr 13, 2013 at 10:29
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$\begingroup$ @Derek: Presumably the group intended is the semidirect product oF elementary Abelian $2$-group of rank $2^{r-1}$ with a Sylow $2$-subgroup of the symmetric group $S_{2^{r-1}},$ or alternatively a Sylow $2$-subgroup of the group of all monomial $2^{r-1} \times 2^{r-1}$ matrices whose only non zero entries are $\pm 1.$ $\endgroup$– Geoff RobinsonCommented Apr 13, 2013 at 12:31
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$\begingroup$ @Geoff: Do you mean with your first description just the $2$-Sylow of the symmetric group $S_{2^r}$? It is for finite fields $\mathbb{F}_q$ with $q=3 \bmod 4$ also the $2$-Sylow of $GL_{2^r}(\mathbb{F}_q)$. $\endgroup$– j.p.Commented Apr 13, 2013 at 14:24
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1$\begingroup$ @jp: But I think you mean a Sylow $2$-subgroup of ${\rm GL}(2^{r-1},q)$) when $q \equiv 3$ (mod 4). $\endgroup$– Geoff RobinsonCommented Apr 13, 2013 at 15:57
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3 Answers
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Since it came up in comments, I will give an answer about the group I believe was intended to be asked about. A monomial matrix is a matrix which has one non-zero entry in each row and one non-zero entry in each column. The monomial $n \times n$ matrices who non-zero entries are all $\pm 1$ form a group, which may be thought of as an abstract group as $Z_{2} \wr S_{n}.$ This matrix group has a normal elementary Abelian subgroup of order $2^{n}$ consisting of all its diagonal matrices. This group has many Sylow $2$-subgroups when $n >2,$ but they are all conjugate within it, so in particular are all isomorphic, and they all contain the normal subgroup consisting of all its diagonal matrices. I believe that the group which was intended to be asked about was such a Sylow $2$-subgroup in the case that $n = 2^{r-1}$.
answered Apr 13, 2013 at 12:49
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You can obtain an embedding of $H$ into ${\rm GL}(2^r,\mathbb{Z})$ as follows:
Given a positive integer $m$, put $$ A_m \ := \ \left( \begin{array}{ll} 0 & 1_m \\\ 1_m & 0 \end{array} \right), $$ where $1_m$ denotes an $m \times m$ unit matrix. Further, given positive integers $m$ and $n$, let $$ B_{m,n} \ := \ \left( \begin{array}{ccc} A_m & & 0 \\\ & \ddots & \\\ 0 & & A_m \\\ \end{array} \right) \ \in \ {\rm GL}(2mn,\mathbb{Z}) $$ be the block diagonal matrix with $n$ blocks $A_m$ on the diagonal and 0 everywhere else. Now we have $H \cong \langle B_{1,2^{r-1}}, B_{2,2^{r-2}}, B_{4,2^{r-3}}, \dots, B_{2^{r-1},1} \rangle < {\rm GL}(2^r,\mathbb{Z})$.
answered Apr 13, 2013 at 9:33
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See:
Leonov, Yu.G.; Yasyns'kyj, V.V. On representation of multiple wreath products of groups ${\mathbb Z}_2$. Visn., Ser. Fiz.-Mat. Nauky, Kyiv. Univ. Im. Tarasa Shevchenka, No. 2, 14-17 (2007).
Summary from Zentralblatt: Representations of multiple wreath products of groups ${\mathbb Z}_2$ by unitriangular matrices over the 2-element field are investigated.
answered Apr 13, 2013 at 9:18