nLab counterexamples in algebra

This page lists counterexamples in algebra.

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

   Thomson's group T.

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 is not a Cohen-Macaulay ring.

   $$k[x,y]/(x^2, x y)$$

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:

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

References

The initial import of counterexamples in this entry was taken from this MO question. See also counterexamples in category theory.