The Laws of Exponents for Groups

# The Laws of Exponents for Groups

If $(G, \cdot)$ is a group, $a \in G$, and $m \in \mathbb{N}$ then we define:

(1)
\begin{align} \quad a^m &= \underbrace{a \cdot a \cdot ... \cdot a}_{m \: \mathrm{terms}} \\ \quad a^{-m} &= \underbrace{a^{-1} \cdot a^{-1} \cdot ... \cdot a^{-1}}_{m \: \mathrm{terms}} \end{align}

By convention, we also define $a^0 = e$ where $e$ is the identity element in $G$.

Just like in regular arithmetic, the laws of exponents also holds on groups $(G, \cdot)$ under their defined operation $\cdot$.

 Proposition 1: Let $(G, \cdot)$ be a group and let $a \in G$. If $m$ is a positive integer then $(a^m)^{-1} = (a^{-1})^m$
• Proof: Let $a \in G$ and let $m$ be a positive integer. Then:
(2)
\begin{align} \quad (a^m)^{-1} = (\underbrace{a \cdot a \cdot ... \cdot a}_{m \: \mathrm{many \: factors}})^{-1} = \underbrace{a^{-1} \cdot a^{-1} \cdot ... \cdot a^{-1}}_{m \: \mathrm{many \: factors}} = (a^{-1})^m \quad \blacksquare \end{align}
 Proposition 2: Let $(G, \cdot)$ be a group and let $a \in G$. If $m$ and $n$ are integers then $a^m \cdot a^n = a^{m+n}$.
• Proof: Let $a \in G$ and let $m$ and $n$ be integers. Then:
(3)
\begin{align} \quad a^m \cdot a^n = (\underbrace{a \cdot a \cdot ... \cdot a}_{m \: \mathrm{many \: factors}}) \cdot (\underbrace{a \cdot a \cdot ... \cdot a}_{n \: \mathrm{many \: factors}}) = \underbrace{a \cdot a \cdot ... \cdot a}_{m+n \: \mathrm{many \: factors}} = a^{m+n} \quad \blacksquare \end{align}
 Proposition 3: Let $(G, \cdot)$ be a group and let $a \in G$. If $m$ and $n$ are integers then $(a^m)^n = a^{mn}$.
• Proof: Let $a \in G$ and let $m$ and $n$ be integers. Then:
(4)
\begin{align} \quad (a^m)^n = (\underbrace{a \cdot a \cdot ... \cdot a}_{m \: \mathrm{many \: factors}})^n = \underbrace{(\underbrace{a \cdot a \cdot ... \cdot a}_{m \: \mathrm{many \: factors}}) \cdot (\underbrace{a \cdot a \cdot ... \cdot a}_{m \: \mathrm{many \: factors}}) \cdot ... \cdot (\underbrace{a \cdot a \cdot ... \cdot a}_{m \: \mathrm{many \: factors}})}_{n \: \mathrm{many \: factors}} = a^{mn} \quad \blacksquare \end{align}