Contents

# Contents

## Group Theory (including quasigroups, semigroups, etc)

1. A non-abelian group, all of whose subgroups are normal:

$Q \coloneqq \langle a, b | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle$
2. A finitely presented, infinite, simple group

3. A group that is not the fundamental group of any 3-manifold.

$\mathbb{Z}^4$
4. Two finite non-isomorphic groups with the same order profile.

$C_4 \times C_4, \qquad C_2 \times \langle a, b, | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle$
5. A quasigroup that is not isomorphic to any loop.

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

$\begin{matrix} * & a & b & c \\ a & a & c & b \\ b & c & b & a \\ c & b & a & c \end{matrix}$
6. A counterexample to the converse of Lagrange's theorem.

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

7. A finite group in which the product of two commutators is not a commutator.

$G = \langle (a c)(b d), (e g)(f h), (i k)(j l), (m o)(n p), (a c)(e g)(i k), (a b)(c d)(m o), (e f)(g h)(m n)(o p), (i j)(k l)\rangle \subseteq S_{16}$
8. 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 $[x^n,y^m]$.

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

A Prüfer group. (The correct theorem is that an Artinian ring is Noetherian.)

## Ring Theory

1. A ring that is right Noetherian but not left Noetherian:

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

2. A ring that is local commutative Noetherian but not Cohen-Macaulay

$k[x,y]/(x^2, x y)$
3. A number ring? that is a principal ideal domain that is not Euclidean.

$\mathbb{Q}(\sqrt{-19})$
4. An epimorphism of rings that is not surjective.

$\mathbb{Z} \to \mathbb{Q}$
5. A ring whose spec has non-open connected components.

$\prod_{n=1}^\infty \mathbb{F}_2$
6. A non-Noetherian ring $A$ such that all local rings on $Spec(A)$ are Noetherian.

$\prod_{n=1}^\infty \mathbb{F}_2$
7. A number field whose ring of integers is Euclidean but not norm-Euclidean.

$\mathbb{Q}(\sqrt{69})$

## Hopf Algebras

1. A non-commutative and non-cocommutative Hopf algebra

\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}

## Homological Algebra

1. An exact sequence that does not split:

$0 \to \mathbb{Z} \stackrel{\times 2}{\to} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0$

## Galois Theory

1. A polynomial, solvable in radicals, 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 rationals, with Galois group cyclic of order $3$.

2. A composition of two normal extensions need not be normal:

$\mathbb{Q} \subset \mathbb{Q}(2^{1/2}) \subset \mathbb{Q}(2^{1/4})$

The initial import of counterexamples in this entry was taken from this MO question.