Subgroups, Cyclic Groups and Lagrange

A subgroup $H \le G$ is a subset that is itself a group under $G$'s operation. The order of an element $a$ is the smallest $k \ge 1$ with $a^k = e$ (in additive groups, the smallest $k$ with $k\cdot a = e$). The powers of a single element form a cyclic subgroup; if one element's powers fill all of $G$, then $G$ is cyclic and that element is a generator.

Lagrange's Theorem: in a finite group, the order of every subgroup divides the order of the group. As a corollary, the order of every element divides $|G|$.

Worked example. In $\mathbb{Z}_6$ (order $6$), take the element $2$: $2,\ 2{+}2{=}4,\ 4{+}2{=}0$. It returns to the identity after $3$ steps, so its order is $3$, and $\{0,2,4\}$ is a subgroup. Note $3 \mid 6$, as Lagrange promises. A generator of $\mathbb{Z}_6$ must have order $6$; $1$ works ($1,2,3,4,5,0$), and so does $5$.