After proving the laws of positive powers, we now extend power notation to all integer exponents. This extension is one of the reasons groups are so useful: every element has an inverse, so negative exponents are meaningful. Once zero and negative exponents are included, the familiar exponent laws continue to hold for a fixed group element. In this lesson, students will learn how to combine positive, zero, and negative powers and why powers of the same element behave like an integer-indexed system inside the group.

Let $(G,\circ)$ be a group with identity element $e$ and let $a\in G$. For $n\in\mathbb{Z}$, the $\textbf{integral power}$ $a^n$ is defined as follows: (i) If $n>0$, then $a^n=\underbrace{a\circ a\circ\cdots\circ a}_{n\text{ times}}$. (ii) If $n=0$, then $a^0=e$. (iii) If $n<0$, then $a^n=(a^{-1})^{-n}$.