Subgroup

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A subgroup is a group contained in another. Specifically, let $H$ and $G$ be groups (with group laws written multiplicatively). 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 also write $H \subseteq G$ 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 $2 \mathbb{Z}/4\mathbb{Z}$ , 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. In a group with identity $e$ , $\{e\}$ is the smallest subgroup.

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 $\{0\}$ .

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