If you find any mistakes, please make a comment! Thank you.

Exhibit symmetric group as a subgroup of a general linear group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.9

Solution: We know that if $F$ is a finite field then $\mathsf{Aut}(F^n) \cong GL_n(F)$. This isomorphism $\zeta$ can be defined as follows: given $\theta \in \mathsf{Aut}(F^n)$, $\zeta(\theta)$ is the matrix in $GL_n(F)$ whose $i$-th row is precisely $\theta(e_i)$. (In particular, $\zeta$ is not canonical since it depends on the choice and order of a basis; here we choose the standard basis.) In Exercises 5.1.7 and 5.1.8, we found an injective group homomorphism $\psi : S_n \rightarrow \mathsf{Aut}(F^n)$. Combining these results we have an injective group homomorphism $\Psi : S_n \rightarrow GL_n(F)$, computed as follows: $$\Psi(\pi) = [e_{\pi(1)}\ \cdots\ e_{\pi(n)}]^T.$$ We can see that each $\Psi(\pi)$ is obtained from the identity matrix by permuting the rows, so that each has a single 1 in each row and column and 0 in all other entries. Thus $S_n$ is identified with a subgroup of $GL_n(F)$ consisting of permutation matrices; moreover, by counting we see that all permutation matrices are represented this way.

Thus we have proven the result for a finite field $F$; we began with this case because it was shown previously that $\mathsf{Aut}(F^n) \cong GL_n(F)$. If this is true for arbitrary fields then the same proof carries over to arbitrary fields $F$; however this will not be proven until later in the text. In the meantime we can convince ourselves that the result holds over arbitrary fields $F$ by noting that, in computing the product of two permutation matrices, we never deal with numbers other than 1 or 0. In particular, the question of whether or not $k = 0$ in $F$ for some integer $k$ never arises, so that all computations hold for arbitrary fields (in which $1 \neq 0$) and in fact the set of permutation matrices over any $F$ is closed under matrix multiplication.


This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.

Leave a Reply

Close Menu