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Show that the group $\mathrm{GL}_2(\mathbb{F}_5)$ has order 480. By defining a suitable homomorphism from $\mathrm{GL}_2(\mathbb{F}_5)$ to another group, which should be specified, show that the order of $\mathrm{SL}_2(\mathbb{F}_5)$ is 120. Find a subgroup of $\mathrm{GL}_2(\mathbb{F}_5)$ of index 2.

For the first part I actually can show the order 480. First I find the order of $\mathrm{M}_2(\mathbb{F}_5)$ , $5^4=625$, then I discard all the singular matrices then I have 480. But find all the cases that the matrices are singular is tedious and not easy to ensure every case is considered in the field. I want to find a simpler and more effective method to approach this.

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    Hint: how many ordered basis has a $\,2-$dimensional vector space over $\,\Bbb F_5:=\Bbb Z/5\Bbb Z\,$ ?2012-12-30
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    http://math.stackexchange.com/questions/68875/how-many-different-basis-exist-for-an-n-dimensional-vector-space-in-mod-2/68885#68885 This answer contains an explanation, but don't divide by n! ...2012-12-30

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