<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\section{Subgroups}

\definition{}
Let $G$ be a group, and let $H$ be a subset of $G$. \par
We say $H$ is a \textit{subgroup} of $G$ if $H$ is also a group
(with the operation $\ast$).

\definition{}
Let $S$ be a subset of $G$. \par
The \textit{group generated by $S$} consists of all elements of $G$ \par
that may be written as a combination of elements in $S$

\vspace{2mm}

We will denote this group as $\langle S \rangle$. \par
Convince yourself that $\langle g \rangle = G$ if $g$ generates $G$.

\problem{}
What is the subgroup generated by $\{7, 8\}$ in $(\znz{15})^\times$? \par
Is this the whole group?

\problem{}
Show that the group generated by $S$ is indeed a group.</pre></body></html>