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\begin{document}

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\section*{counterexamples in algebra}

This page lists counterexamples in [[algebra]].

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{group_theory_including_quasigroups_semigroups_etc}{Group Theory (including quasigroups, semigroups, etc)}\dotfill \pageref*{group_theory_including_quasigroups_semigroups_etc} \linebreak
\noindent\hyperlink{ring_theory}{Ring Theory}\dotfill \pageref*{ring_theory} \linebreak
\noindent\hyperlink{hopf_algebras}{Hopf Algebras}\dotfill \pageref*{hopf_algebras} \linebreak
\noindent\hyperlink{homological_algebra}{Homological Algebra}\dotfill \pageref*{homological_algebra} \linebreak
\noindent\hyperlink{galois_theory}{Galois Theory}\dotfill \pageref*{galois_theory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{group_theory_including_quasigroups_semigroups_etc}{}\subsection*{{Group Theory (including quasigroups, semigroups, etc)}}\label{group_theory_including_quasigroups_semigroups_etc}

\begin{enumerate}%
\item A non-[[abelian group|abelian]] [[group]], all of whose [[subgroup]]s are [[normal subgroup|normal]]:

\begin{displaymath}
Q \coloneqq \langle a, b | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
\end{displaymath}

\item A [[finitely presented group|finitely presented]], infinite, [[simple group]]

[[Thomson's group]] T.


\item A [[group]] that is not the [[fundamental group]] of any [[3-manifold]].

\begin{displaymath}
\mathbb{Z}^4
\end{displaymath}

\item Two [[finite group|finite]] non-[[isomorphism|isomorphic]] groups with the same [[order profile]].

\begin{displaymath}
C_4 \times C_4, \qquad C_2 \times \langle a, b, | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
\end{displaymath}

\item A [[quasigroup]] that is not isomorphic to any [[loop (algebra)|loop]].

$\{a, b, c\}$ with multiplication table:

\begin{displaymath}
\begin{matrix}
* & a & b & c \\
a & a & c & b \\
b & c & b & a \\
c & b & a & c
\end{matrix}
\end{displaymath}

\item A counterexample to the converse of [[Lagrange's theorem]].

The [[alternating group]] $A_4$ has order $12$ but no [[subgroup]] of order $6$.


\item A [[finite group]] in which the product of two [[commutator]]s is not a commutator.

\begin{displaymath}
G = \langle (a c)(b d), (e g)(f h), (i k)(j l), (m o)(n p), (a c)(e g)(i k), (a b)(c d)(m o), (e f)(g h)(m n)(o p), (i j)(k l)\rangle \subseteq S_{16}
\end{displaymath}

\item A [[finitely generated group]] with a non-finitely generated subgroup.

The free group on two generators $x$ and $y$ has commutator subgroup freely generated by $[x^n,y^m]$.


\item An Artinian but not Noetherian $\mathbb{Z}$-module.

A [[Prüfer group]]. (The correct theorem is that an [[Artinian ring|Artinian]] \emph{[[ring]]} is [[Noetherian ring|Noetherian]].)



\end{enumerate}
\hypertarget{ring_theory}{}\subsection*{{Ring Theory}}\label{ring_theory}

\begin{enumerate}%
\item A [[ring]] that is right [[Noetherian ring|Noetherian]] but not left Noetherian:

Matrices of the form $\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$ where $a \in \mathbb{Z}$ and $b,c \in \mathbb{Q}$.


\item A ring that is local commutative Noetherian but not Cohen-Macaulay

\begin{displaymath}
k[x,y]/(x^2, x y)
\end{displaymath}

\item A [[number ring]] that is a [[principal ideal domain]] that is not Euclidean.

\begin{displaymath}
\mathbb{Q}(\sqrt{-19})
\end{displaymath}

\item An [[epimorphism]] of [[rings]] that is not [[surjective]].

\begin{displaymath}
\mathbb{Z} \to \mathbb{Q}
\end{displaymath}

\item A [[ring]] whose [[spectrum|spec]] has non-[[open subset|open]] [[connected]] components.

\begin{displaymath}
\prod_{n=1}^\infty \mathbb{F}_2
\end{displaymath}

\item A non-[[Noetherian ring|Noetherian]] [[ring]] $A$ such that all local rings on $Spec(A)$ are Noetherian.

\begin{displaymath}
\prod_{n=1}^\infty \mathbb{F}_2
\end{displaymath}

\item A [[number field]] whose [[ring of integers]] is Euclidean but not norm-Euclidean.

\begin{displaymath}
\mathbb{Q}(\sqrt{69})
\end{displaymath}


\end{enumerate}
\hypertarget{hopf_algebras}{}\subsection*{{Hopf Algebras}}\label{hopf_algebras}

\begin{enumerate}%
\item A non-commutative and non-cocommutative [[Hopf algebra]]

\begin{displaymath}
\begin{aligned}
H &\coloneqq &\langle x, g | g^2 = 1, x^2 = 0, g x g = -x\rangle \\
\Delta(g) &= &g \otimes g, \\
\Delta(x) &= &x \otimes 1 + g \otimes x, \\
\epsilon(g) &=& 1, \\
\epsilon(x) &=& 0, \\
S(g) &= &g, \\
S(x) &= &- g x
\end{aligned}
\end{displaymath}


\end{enumerate}
\hypertarget{homological_algebra}{}\subsection*{{Homological Algebra}}\label{homological_algebra}

\begin{enumerate}%
\item An [[exact sequence]] that does not [[split sequence|split]]:

\begin{displaymath}
0 \to \mathbb{Z} \stackrel{\times 2}{\to} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0
\end{displaymath}


\end{enumerate}
\hypertarget{galois_theory}{}\subsection*{{Galois Theory}}\label{galois_theory}

\begin{enumerate}%
\item A [[polynomial]], solvable in [[radical]]s, whose [[splitting field]] is not a [[radical extension]] of $\mathbb{Q}$.

Take any cyclic cubic; that is, any cubic with rational coefficients, irreducible over the [[rational number|rational]]s, with [[Galois group]] cyclic of order $3$.


\item A composition of two normal extensions need not be normal:



\end{enumerate}
\begin{displaymath}
\mathbb{Q} \subset \mathbb{Q}(2^{1/2}) \subset \mathbb{Q}(2^{1/4})
\end{displaymath}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The initial import of counterexamples in this entry was taken from \href{http://mathoverflow.net/questions/29006/counterexamples-in-algebra}{this MO question}.



\end{document}