This page lists counterexamples in algebra.
A non-abelian group, all of whose subgroups are normal:
A finitely presented, infinite, simple group
A group that is not the fundamental group of any 3-manifold.
Two finite non-isomorphic groups with the same order profile.
A quasigroup that is not isomorphic to any loop.
$\{a, b, c\}$ with multiplication table:
A counterexample to the converse of Lagrange's theorem.
The alternating group $A_4$ has order $12$ but no subgroup of order $6$.
A finite group in which the product of two commutators is not a commutator.
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 $[x^n,y^m]$.
An Artinian but not Noetherian $\mathbb{Z}$-module.
A Prüfer group. (The correct theorem is that an Artinian ring is Noetherian.)
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 $b,c \in \mathbb{Q}$.
A ring that is local commutative Noetherian but not Cohen-Macaulay
A number ring? that is a principal ideal domain that is not Euclidean.
An epimorphism of rings that is not surjective.
A ring whose spec has non-open connected components.
A non-Noetherian ring $A$ such that all local rings on $Spec(A)$ are Noetherian.
A number field whose ring of integers is Euclidean but not norm-Euclidean.
A non-commutative and non-cocommutative Hopf algebra
An exact sequence that does not split:
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 $3$.
A composition of two normal extensions need not be normal:
The initial import of counterexamples in this entry was taken from this MO question. See also counterexamples in category theory.
Last revised on March 14, 2021 at 13:53:34. See the history of this page for a list of all contributions to it.