Please Login to access more options.

Problem 71 (The Special Linear Group Is A Subgroup Of The General Linear Group)

Prove that $\text{SL}(m,\mathbb{Z}_n)$ is a subgroup of $\text{GL}(m,\mathbb{Z}_n)$. Why is the set of matrices whose determinant equals 2 not a subgroup of the general linear group?

Solution

We will prove that $\text{SL}(m,\mathbb{Z}_{n})$ is a subgroup of $\text{GL}(m,\mathbb{Z}_{n})$ using the subgroup test. We must first show that $\text{SL}(m,\mathbb{Z}_n)$ is a nonempty subset of $\text{GL}(m,\mathbb{Z}_n)$. Recall the definition $\text{SL}(m,\mathbb{Z}_n)=\{a\in \text{GL}(m,\mathbb{Z}_n)|\text{det}(a)=1\}$. This means that we know that $\text{SL}(m,\mathbb{Z}_n)$ is a subset of $\text{GL}(m,\mathbb{Z}_n)$. Now consider the identity matrix $I_m\in\text{GL}(m,\mathbb{Z}_n)$. We know that $I_m\in\text{SL}(m,\mathbb{Z}_n)$, since $\text{det}(I_m)=1$. Thus, $\text{SL}(m,\mathbb{Z}_n)$ is a nonempty subset of $\text{GL}(m,\mathbb{Z}_n)$.

We will now show that $\text{SL}(m,\mathbb{Z}_n)$ is closed under taking inverses. Let $a\in\text{SL}(m,\mathbb{Z}_n)$. Since $a\in\text{GL}(m,\mathbb{Z}_n)$, then $a^{-1}\in\text{GL}(m,\mathbb{Z}_n)$. Now by MATH 341 we know $\text{det}(a^{-1}a)=\text{det}(a^{-1})\text{det}(a)=\text{det}(a^{-1})(1)=\text{det}(a^{-1})$. This means $\text{det}(a^{-1})=\text{det}(a^{-1}a)=\text{det}(I_m)=1$. Therefore, $\text{SL}(m,\mathbb{Z}_n)$ is closed under taking inverses.

Let $a,b\in\text{SL}(m,\mathbb{Z}_n)$. We will now compute $\text{det}(ab)=\text{det}(a)\text{det}(b)=(1)(1)=1$. This means that $\text{SL}(m,\mathbb{Z}_n)$ is closed under the binary operation of $\text{GL}(m,\mathbb{Z}_n)$.

Hence, by the subgroup test $\text{SL}(m,\mathbb{Z}_n)$ is a subgroup of $\text{GL}(m,\mathbb{Z}_n)$. In addition, we know that the set of matrices whose determinant equals 2 is not a subgroup of the general linear group because this set does not contain the identity matrix $I_m\in\text{GL}(m,\mathbb{Z}_n)$ whose determinant is equal to 1.

Tags

Change these as needed.

  • When you are ready to submit this written work for grading, add the phrase [[!Submit]] to your page. This will tell me that you have completed the page (it's past rough draft form, and you believe it is in final draft form). Don't type [[!Submit]] on a rough draft.
  • If I put [[!NeedsWork]] on your page, then your job is to review what I've written, address any comments made, and then delete all the comments I made. When you have finished reviewing your work, leave [[!NeedsWork]] on your page and type [[!Submit]]. (Both tags will show up). This tells me you have addressed the comments.
  • I'll mark your work with [[!Complete]] after you have made appropriate revisions.