From 98265949d89662d6711fd00f57c9325a4d77d5e4 Mon Sep 17 00:00:00 2001
From: Robert <9291792+bushshrub@users.noreply.github.com>
Date: Thu, 14 Nov 2024 11:44:38 -0500
Subject: [PATCH] add some remarks on the finite abelian group classification
 theorem

---
 chapter-classification-of-finite-abelian-groups.tex | 11 +++++++++++
 chapter-group-actions.tex                           |  3 ---
 2 files changed, 11 insertions(+), 3 deletions(-)

diff --git a/chapter-classification-of-finite-abelian-groups.tex b/chapter-classification-of-finite-abelian-groups.tex
index 219e3cd..610ba3c 100644
--- a/chapter-classification-of-finite-abelian-groups.tex
+++ b/chapter-classification-of-finite-abelian-groups.tex
@@ -167,6 +167,17 @@ \section{Classification of finite abelian groups}
 \end{proof}
 
 
+We end off this chapter with some closing remarks. Firstly, the group $\gen a$
+in \cref{lem:lemma-prime-power-order-abelian-groups-factorization} would be a
+Sylow $p$-subgroup of $G$.
+
+Secondly, this theorem can be derived as a corollary of a more general theorem,
+the classfiication of finitely generated abelian groups. This theorem can be
+derived as a corollary of a even more general theorem, the classification of
+finitely generated modules over a principal ideal domain. We will take up these
+theorems in the future, but for now these facts are just interesting to note.
+
+
 \subsection{Exercises and Problems}
 
 \begin{exercise}[Subgroups of finite abelian groups]
diff --git a/chapter-group-actions.tex b/chapter-group-actions.tex
index b58f647..6d00ca0 100644
--- a/chapter-group-actions.tex
+++ b/chapter-group-actions.tex
@@ -700,9 +700,6 @@ \section{The class equation and Sylow theorems}
 \end{proof}
 
 
-
-
-
 \subsection{Problems and exercises}
 \begin{exercise}[Conjugate of Sylow $p$-subgroup is a Sylow $p$-subgroup]
 \label{ex:conjugate-of-sylow-subgroup}