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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:

   $$Q≔⟨a,b\mid a^4=1,a^2=b^2,ab=b{a}^3⟩$$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.

   $${\mathbb{Z}}^4$$\mathbb{Z}^4

4. Two finite non-isomorphic groups with the same order profile.

   $$C_4\times C_4,C_2\times ⟨a,b,\mid a^4=1,a^2=b^2,ab=b{a}^3⟩$$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:

   $$\begin{array}{cccc}*& a& b& c\\ a& a& c& b\\ b& c& b& a\\ c& b& a& c\end{array}$$\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.

   $$G=⟨(ac)(bd),(eg)(fh),(ik)(jl),(mo)(np),(ac)(eg)(ik),(ab)(cd)(mo),(ef)(gh)(mn)(op),(ij)(kl)⟩\subseteq S_{16}$$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 $\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]$ where $a\in \mathbb{Z}$ and $b,c\in \mathbb{Q}$.

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

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

3. A number ring? that is a principal ideal domain that is not Euclidean.

   $$\mathbb{Q}(\sqrt{-19})$$\mathbb{Q}(\sqrt{-19})

4. An epimorphism of rings that is not surjective.

   $$\mathbb{Z}\to \mathbb{Q}$$\mathbb{Z} \to \mathbb{Q}

5. A ring whose spec has non-open connected components.

   $$\prod _{n=1}^{\mathrm{\infty }}{𝔽}_2$$\prod_{n=1}^\infty \mathbb{F}_2

6. A non-Noetherian ring $A$ such that all local rings on $\mathrm{Spec}(A)$ are Noetherian.

   $$\prod _{n=1}^{\mathrm{\infty }}{𝔽}_2$$\prod_{n=1}^\infty \mathbb{F}_2

7. A number field whose ring of integers is Euclidean but not norm-Euclidean.

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

Hopf Algebras

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

   $$\begin{array}{rlr}H& ≔& ⟨x,g\mid g^2=1,x^2=0,gxg=-x⟩\\ \Delta (g)& =& g\otimes g,\\ \Delta (x)& =& x\otimes 1+g\otimes x,\\ ϵ(g)& =& 1,\\ ϵ(x)& =& 0,\\ S(g)& =& g,\\ S(x)& =& -gx\end{array}$$\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:

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

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