> [!NOTE] Definition (Exponent) > Let $G$ be a [[Groups|group]], we say that $G$ has exponent $n\in \mathbb{N}^+$ if for all $g\in G,$ $g^n=\text{id}$ . **Note**: By Lagrange, if $G$ is finite then $\exp(G)$ divides $\text{ord}(G)$ and so is finite. # Properties