23 lines
679 B
TeX
23 lines
679 B
TeX
|
\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.
|