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Consider a finite field $K$ with $q$ elements. For each $0\leq k\leq n$, there are ${n \brack k}_q$ subspaces of $K^n$ of dimension $k$. I want to count cosets. Let $A$ be a subspace of $K^n$ of dimension $k$. By simple group theory, there are $[K^n:A]=q^n/|A|$ cosets of the form $x+A$ $(x\in K^n)$. I am not sure how to combine this with ${n \brack k}_q$ to get the number of cosets of dimension $k$. Any suggestions?

user831160
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  • Hi Ruan, What is the problem with multiplying ${n \brack k}_q$ by $\frac{q^n}{|A|}$? – Amirhossein Oct 01 '20 at 15:27
  • @Amirhossein Well in this https://math.stackexchange.com/questions/315762/calculating-number-of-lines-and-planes-in-affine-space-over-finite-field/1677280#1677280 answer they multiply the gaussian coefficient by $q^{n-k}$. I thought I could show that $|A|$ is $q^{k}$ to conclude the same thing. But I'm not sure – user831160 Oct 01 '20 at 16:22
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    Indeed, $|A|=q^k$, since the dimension of $A$ is $k$, so $q^n/|A|=q^{n-k}$. – Mike Earnest Oct 01 '20 at 17:19
  • @Ruan, The thing that you claimed is correct and simple. Plus by multiplying it with that number you would get the number of cosets. – Amirhossein Oct 01 '20 at 17:52

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