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Subgroups

A group may contain other groups within it. For example, the group $\mathbb{Q}$ (with respect to addition) contains the group $\mathbb{Z}$ , in the sense that ${\mathbb{Z}}\subseteq {\mathbb{Q}}$ and that the addition in $\mathbb{Z}$ is the same as the addition in $\mathbb{Q}$ restricted to $\mathbb{Z}$ .

$\begin{de} Let $G$\ be a group. A non-empty set $H\subseteq G$is a {\em subgroup... ...it forms a group under the same operation; we denote this by $H\leq G$. \end{de}$

$\begin{ex} ${\mathbb{Z}}\leq {\mathbb{Q}}\leq {\mathbb{R}}\leq {\mathbb{C}}$. \end{ex}$

$\begin{ex} The multiplicative group ${\mathbb{Q}}\setminus\{0\}$is not a subgrou... ...arly, ${\mathbb{Z}}_m$\ is not a subgroup of ${\mathbb{Z}}_n$\ ($m<n$). \end{ex}$

$\begin{ex} Every group $G$\ has $\{e\}$\ as a subgroup; this is called the {\em ... ...oup different from $\{e\}$\ and $G$\ is called a {\em proper} subgroup. \end{ex}$

On the face of it, to check whether a subset of a group is a subgroup, we have to check the four axioms for groups. In fact, this can be reduced to checking two conditions:

$\begin{thm} Let $H$\ be a non-empty subset of a group $G$. Then $H$\ is a subgro... ... and only if $H$\ is closed under multiplication and taking inverses). \end{thm}$

$\begin{proof} ($\Rightarrow$) This follows immediately from the definition. \par... ...lly, the operation is associative, because it is associative in $G$. \end{proof}$

$\begin{ex} The set $GL(n,{\mathbb{R}})$\ of all $n\times n$\ invertible matrices... ...s. \par Note that ${\mathbb{Q}}\setminus\{0\} \leq GL(2,{\mathbb{Q}})$. \end{ex}$

$\begin{ex} $\left\{ \left(\begin{array}{cc}1&n\\ 0&1\end{array}\right)\st n\in{\mathbb{Z}}\right\} \leq GL(2,{\mathbb{Q}})$. \end{ex}$

$\begin{ex} $\{0,2,4\}$\ is a subgroup of ${\mathbb{Z}}_6$\ under $+$. \end{ex}$

$\begin{exc} Let $G = {\mathbb{Q}}\setminus\{0\}$\ under multiplication. Show that $H = \{2^n : n \in \mathbb{Z}\}$\ is a subgroup. \end{exc}$

Edmund F Robertson

11 September 2006