After studying semigroup conditions for groups, we return to groups and develop the language of powers. Powers are repeated products of a single group element. This notation is essential because many later concepts, such as order of an element, cyclic group, generator, and roots of identity, are expressed using powers. In ordinary arithmetic, powers are familiar, but in an abstract group they must be defined through the group operation. In this lesson, students will learn positive powers, zero power, and negative powers of an element in a group.

Let $(G,\circ)$ be a group with identity element $e$ and let $a\in G$. For $n\in\mathbb{N}$, the $\textbf{positive power}$ $a^n$ is defined by $$\begin{equation*} a^n=\underbrace{a\circ a\circ\cdots\circ a}_{n\text{ times}}. \end{equation*}$$ Also, $$\begin{equation*} a^0=e. \end{equation*}$$