wreath product and matrix presentation 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
 A: 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 $n = 2^{r-1}$.
A: 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 $H \cong \langle B_{1,2^{r-1}}, B_{2,2^{r-2}}, B_{4,2^{r-3}},
\dots, B_{2^{r-1},1} \rangle < {\rm GL}(2^r,\mathbb{Z})$.
A: 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.
