diff options
Diffstat (limited to 'library/algebra')
| -rw-r--r-- | library/algebra/group.tex | 12 | ||||
| -rw-r--r-- | library/algebra/monoid.tex | 2 |
2 files changed, 12 insertions, 2 deletions
diff --git a/library/algebra/group.tex b/library/algebra/group.tex index a79bd2f..449bacb 100644 --- a/library/algebra/group.tex +++ b/library/algebra/group.tex @@ -1,5 +1,5 @@ \import{algebra/monoid.tex} -\section{Group} +\section{Group}\label{form_sec_group} \begin{struct}\label{group} A group $G$ is a monoid such that @@ -80,3 +80,13 @@ \begin{definition}\label{group_automorphism} Let $f$ be a function. $f$ is a group-automorphism iff $G$ is a group and $\dom{f}=G$ and $\ran{f}=G$. \end{definition} + +\begin{definition}\label{trivial_group} + $G$ is the trivial group iff $G$ is a group and $\{\neutral[G]\}=G$. +\end{definition} + +\begin{theorem}\label{trivial_implies_abelian} + Let $G$ be a group. + Suppose $G$ is the trivial group. + Then $G$ is an abelian group. +\end{theorem} diff --git a/library/algebra/monoid.tex b/library/algebra/monoid.tex index 06fcb50..3249a93 100644 --- a/library/algebra/monoid.tex +++ b/library/algebra/monoid.tex @@ -1,5 +1,5 @@ \import{algebra/semigroup.tex} -\section{Monoid} +\section{Monoid}\label{form_sec_monoid} \begin{struct}\label{monoid} A monoid $A$ is a semigroup equipped with |