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Can we compute the cardinality of the linear group $GL_n({\mathbb Z}/{p \mathbb Z})$?

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marked as duplicate by quid, Thomas, Batominovski, Daniel Fischer Oct 17 '15 at 21:19

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Yes we can! Representing the elements of this group in a fixed basis as matrices, you can reason as follows:

  1. How many options are there for the first column?
  2. After having selected the first column, how many options are there for the second one? It is here that you should use the fact that your matrix has to be invertible, and so the columns have to be linearly independent.
  3. Repeat column by column.
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  • $\begingroup$ The coefficients of the linear combinations are from Z/pZ? $\endgroup$ – alexb Oct 17 '15 at 16:55
  • $\begingroup$ Yes, since you are considering vectors in the space $(\mathbb Z / p \mathbb Z)^n$ over $\mathbb Z / p \mathbb Z$. $\endgroup$ – darko Oct 17 '15 at 17:54
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Using the notion of linear combination, we can easily calculate the cardinality.For the first column we have $p^n-1$,the second,$p^n-p$,……,the last,$p^n-p^{n-1}$.Simply multiply them.

As a further question:one may calculate the cardinality of Sylow-p subgroup of $G=GL_n(\mathbb Z/\mathbb p\mathbb Z)$.Apply the conjugate action to the set of all Sylow-p subgroups.Then the action is transitive.Firstly, you should get one Sylow-p subgroup(which is easy to get,considering the upper triangular matrix with diagonal elements being 1).Secondly ,find the stable subgroup,in this case,the normalizer.As an exercise,you should test that the upper triangular matrix is the normalizer.As a result,the cardinality of Sylow-p subgroup is $\frac{\prod_{k=1}^{n}{p^k-1}}{(p-1)^n}$

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