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{\hfill{Fall 2021} \smallskip %
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{\LARGE Introduction to Algebraic Structures\\} 

{\LARGE Problem Set 3\\}
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{\LARGE Abelian groups.}
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\noindent{\fontsize{10}{12}\selectfont \href{https://en.wikipedia.org/wiki/Finitely_generated_abelian_group#Primary_decomposition}{Fundamental Theorem of Finitely Generated Abelian Groups}: Theorem 11.6.5 of M. Kilp's textbook ``Algebra I'' (2005), Theorem 3 in section 5.2 of Dummit\&Foote's ``Abstract Algebra'', or see the link.

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\noindent 1b. Prove that any monoid satisfying the identity $aba = b$ is an Abelian group. 

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\noindent 2. Find the maximal order of the elements of $\Z_{8}\times\Z_{12}$ and a concrete element of that order.

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\noindent 3${^*}$. Prove that any group of order 2021 is Abelian.

\hfill\vskip-20pt\begin{turn}{180}{\tiny{\it Hint.} Cauchy's theorem.}\hfill\end{turn}

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\noindent 3b. Prove that any Abelian group of order 2021 is cyclic. No using the FTFGAG or some variant of the Chinese Remainder Theorem, group theory only. 

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\noindent 4. Prove that a \emph{finite} Abelian $p$-group is generated by its elements of maximal order. 

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\noindent 5. Find the orders of all elements of the group $U(\Z_{27})$ of all invertible elements of $\Z_{27}$ and decompose $U(\Z_{27})$ into the (internal) direct sum of its $p$-components.

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\noindent 6. Prove that the (internal) sum of all divisible subgroups of an Abelian group is divisible.

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\noindent 7. Let $A$ be a finite Abelian group and $B\leq A$. Prove that $A$ has a subgroup isomorphic to $A/B$.   

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\noindent 8. Find the product of all the elements of a finite Abelian group. 
\\\hfill\vskip-20pt\begin{turn}{180}{\tiny{\it Hint.} Start by finding the order of the product.}\hfill\end{turn}
 
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\noindent 9${^*}$. Prove that the (primary) decomposition of a finitely generated Abelian group (i.e. FTFGAG) is unique up to isomorphism (between addends). 

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\noindent 10${^{*}}$. Prove that there are no uncountably infinite Abelian groups such that \emph{all} their proper subgroups are finite. 

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