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## Compute the index of the special linear group in the general linear group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.1

Let $F$ be a finite field of order $q$ and let $n$ be a positive integer. Prove that $$[GL_n(F):SL_n(F)] = q-1.$$

Solution: We proved in Exercise 3.1.35 that $$GL_n(F)/SL_n(F) \cong F^\times,$$ where $F^\times$ denotes the set of all elements in $F$ which have a multiplicative inverse. By definition, this is all elements except $0$. So the index is $q-1$.