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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 $H$ of a group $G$ 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