Subgroup

A subgroup is a group contained in another. Specifically, let $H$ and $G$ be groups. We say that $H$ is a subgroup of $G$ if the elements of $H$ constitute a subset of the set of elements of $G$ and the group law on $H$ agrees with group law on $G$ where both are defined. We may denote this by $H \subseteq G$ or $H \le G$.

We say that $H$ is a proper subgroup of $G$ if $H \neq G$.

Examples

In the additive group $\mathbb{Z}/4\mathbb{Z}$, shown below, $$\begin{array}{c|cccc} &0&1&2&3 \\\hline 0&0&1&2&3 \\ 1&1&2&3&0 \\ 2&2&3&0&1 \\ 3&3&0&1&2 \end{array}$$ there are three subgroups : the group itself, $\{ 0 \}$, and the group $2 \mathbb{Z}/4\mathbb{Z} = \{0, 2\}$, shown below. This last subgroup is isomorphic to the additive group $\mathbb{Z}/2\mathbb{Z}$. $$\begin{array}{c|cc} & 0& 2 \\\hline 0&0&2 \\ 2&2&0 \end{array}$$

Every group is the largest subgroup of itself. The set consisting of the identity element of a group is the smallest subgroup of that group.

In a group $G$, the intersection of a family of subgroups of $G$ is a subgroup of $G$. Thus for any collection $X$ of elements of $G$, there exists a smallest subgroup containing these elements. This is called the subgroup generated by $X$.

In the additive group $\mathbb{Z}$, all subgroups are of the form $n \mathbb{Z}$ for some integer $n$. In particular, for $n=1$ we have the integers themselves and for $n=0$ we have $\{0\}$.