Basic Group Theory

 

Multiplicative inverses modulo $m$

The multiplicative group of integers modulo $m$ is defined as: \begin{align} \Z_m^* = \{a\ |\ \gcd(a,m) = 1\} \end{align} But why? This is because Euler’s theorem says that: \begin{align} \gcd(a,m) = 1\Rightarrow a^{\phi(m)} = 1 \end{align} This in turn, implies that every element in $\Z_m^*$ has an inverse, since: \begin{align} a\cdot a^{\phi(m) - 1} &= 1 \end{align} Thus, for a prime $p$, all elements in $\Z_p^* = \{1,2,\dots, p-1\}$ have inverses. Specifically, the inverse of $a \in \Z_p^*$ is $a^{p-2}$.