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Let $S$ be the set of matrices defined inductively as follows:

  • The scalar $1 \in S$.
  • $M \in S$ if $M$ is a $2 \times 2$ block diagonal matrix where the blocks are recursively in $S$.
  • $M \in S$ if $M$ is a $2 \times 2$ block anti-diagonal matrix where the blocks are recursively in $S$.

Are there any special names / properties of $S$?

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Why did you add the group theory tag? In other words, what is the motivation of this question? Just curious... –  t.b. May 28 '11 at 8:00
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I'm not aware of any names, but maybe you can clarify what you're looking for in terms of properties. (They have determinant $\pm 1$, but surely this much is obvious. Another simple property is the product of any two elements of $S$ (of the same size) is block diagonal.)

The matrices in $S$ of size $2^n \times 2^n$ will be permutation matrices for permutations which one might call "bipartite," in the sense that it is either a product of a permutation on $\{ 1, \ldots, 2^{n-1} \}$ with one on $\{ 2^{n-1}+1 , \ldots, 2^{n} \}$, or the composition of such a product with the permutation switching $i$ and $2^{n-1}+i$. (In the latter case, the corresponding graph will be bipartite.) Furthermore, each of the subpermutations is again of this "bipartite" type.

Consequently for each dimension $2^n$, the matrices in $S$ form a subgroup of $S_{2^n}$ (in general not abelian), and the order is easy to count: $2^{2^0+2^1+\cdots+2^{n-1}}=2^{2^n-1}$. Examining the first few cases, suggest that these matrices in fact give a 2-Sylow subgroup of $S_{2^n}$.

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If you have a group of matrices G, and a group of permutations H of degree n, then the wreath product of G with H is the group of n×n block matrices whose nonzero blocks are from G, and are in the positions described by H: the block in the i th row is in the π( i )th column. In this case, H = { (), (1,2) }, and G becomes an iterated wreath product of copies of H, so indeed, a Sylow 2-subgroup of Sym(2^n). –  Jack Schmidt May 29 '11 at 7:16
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