WebDec 15, 2014 · $\begingroup$ Continuing...user6312's observation and the multiplication principle of counting will get you only the answer to 1). The easiest way to do 2) is to use 1) and a little group theory, if you know some. The determinant function $\det \colon \mathrm{GL}_n (\mathbb{F}_3) \rightarrow \mathbb{F}_3^{\times}$ is a group … WebThere's a fibration. G L ( n, C) → G L ( n + 1, C) → C n + 1 ∖ { 0 } By Gram-Schmidt, this fibration is fibre homotopy-equivalent to. U n → U n + 1 → S 2 n + 1. given by only remembering the 1st vector in the matrix, just as in your S O ( n) example. The stable homotopy groups of the unitary groups are known. Google "Bott Periodicity".

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