Fixed DS Lec27

Author: Abhinav271828

_posts/2020-11-19-ds-lecture-27.md

@@ -29,12 +29,12 @@ We will prove Langrange's Theorem and three of its corollaries.
 #### Theorem.
 The order of a finite group $G$ is divisible by the order of its subgroup $H$.
 
-**Proof.** Let $|G| = n$ and $|H| = m$. We know that all left cosets of $H$ have $m$ elements, and form a partition of $G$. Since $G$ is finite, the number of these cosets is also finite; call it $k$. Hence $n = mk$ and therefore, $m|n$, QED.
+**Proof.** Let $n$ be the order of $G$ and $m$ the order of H. We know that all left cosets of $H$ have $m$ elements, and form a partition of $G$. Since $G$ is finite, the number of these cosets is also finite; call it $k$. Hence $n = mk$ and therefore, $m$ divides $n$, QED.
 
 #### Corollary 1.
 The order of $G$ is divisible by the index $k$ [number of distinct cosets] of $H$.
 
-**Proof.** It has been shown that $n = mk$; from this it follows directly that $k | n$.
+**Proof.** It has been shown that $n = mk$; from this it follows directly that $k$ divides $n$.
 
 #### Corollary 2.
 The order of $G$ is divisble by the orders of all $g \in G$.