Add pairings initial notes

abstract-algebra-charles-pinter-notes.tex

@@ -179,7 +179,7 @@ Every subgroup of a cyclic group is cyclic.
 \end{theorem}
 
 \begin{theorem}[Lagrange's theorem]
-    Let $G$ be a finite group, and $H$ any subgroup of $G$. The order of $G$ is a multiple of the order of $H$.
+    Let $G$ be a finite group, and $H$ any subgroup of $G$. The order of $G$ is a multiple of the order of $H$. $|H|$ divides $|G|$.
 \end{theorem}
     Lagrange's theorem can be easily seen by the facts that:
     \begin{enumerate}[i.]
@@ -187,7 +187,6 @@ Every subgroup of a cyclic group is cyclic.
 	\item $|Ha| = |H|$ (each coset has the same order as H). 
     \end{enumerate}
 
-
 By consequence,
 \begin{theorem}
     If $G$ is a group with a prime number $p$ of elements, then $G$ is a cyclic group. Furthermore, any element $a \neq e$ in $G$ is a generator of $G$.
@@ -371,6 +370,18 @@ From the last two theorems: every integer $m$ can be factored into primes, and t
     $$a^{p-1} \equiv 1 \pmod p, \forall a \not\equiv 0 \pmod p$$
     \\
     So, by taking $a^{p-2} \cdot a \equiv 1 \pmod p$, where $a^{p-2} \equiv a^{-1} \pmod p$ (the inverse modulo p), we see that $a^p \equiv a \pmod p, \forall a \in \mathbb{Z}$, so $a^p - a$ is a multiple of $p$.
+
+    ~\\\emph{Relation to Lagrange's theorem:}\\
+    Let $G = \mathbb{Z}_p$, and let $H$ be the multiplicative subgroup of $G$ generated by $a$ (ie. $H = \{ 1, a, a^2, \ldots \}$). The order of $H$ ($h = |H|$), is also the order of $a$ (ie. smallest $n>1$ s.t. $a^n=1~mod~p$).
+
+    By Lagrange's theorem, $h~|~|G| = p - 1$, so $p-1 = h \cdot m$, thus
+    $$
+    a^{p-1} = (a^h)^m \equiv 1^m \equiv 1~mod~p
+    $$
+
+    ~\\\emph{Another perspective:}\\
+    We have $a^p \equiv a \pmod{p}$, by dividing by $a$ on both sides, we obtain $a^{p-1} \equiv 1 \pmod{p}$.
+
 \end{theorem}
 
 \begin{theorem}[Euler's $\phi$ function]