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Following my previous question (not necessary to refer to), let the number of automorphisms of an $n$-dimensional vector space be $k$. Then am I right in thinking that the number of $m$-dimensional subspaces is simply ${n\choose m}k$? Thanks again.

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This is obviously false: take $m=n$.

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    @Shaun: an $n$-dimensional space has precisely one $n$-dimensional subspace.2011-09-14