Started QG handout

Advanced/Quotient Groups/parts/2 subgroups.tex

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+\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.