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counterexamples in algebra

This page lists counterexamples in algebra.

Contents

Group Theory (including quasigroups, semigroups, etc)

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

    Qa,ba 4=1,a 2=b 2,ab=ba 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.

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

    C 4×C 4,C 2×a,b,a 4=1,a 2=b 2,ab=ba 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}\{a, b, c\} with multiplication table:

    * a b c a a c b b c b a c b a c \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 4A_4 has order 1212 but no subgroup of order 66.

  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)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 xx and yy has commutator subgroup freely generated by [x n,y m][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 [a b 0 c]\begin{bmatrix} a & b \\ 0 & c \end{bmatrix} where aa \in \mathbb{Z} and b,cb,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.

    (19) \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.

    n=1 𝔽 2 \prod_{n=1}^\infty \mathbb{F}_2
  6. A non-Noetherian ring AA such that all local rings on Spec(A)Spec(A) are Noetherian.

    n=1 𝔽 2 \prod_{n=1}^\infty \mathbb{F}_2
  7. A number field whose ring of integers is Euclidean but not norm-Euclidean.

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

Hopf Algebras

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

    H x,gg 2=1,x 2=0,gxg=x Δ(g) = gg, Δ(x) = x1+gx, ϵ(g) = 1, ϵ(x) = 0, S(g) = g, S(x) = gx \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:

    0×2/20 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 33.

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

(2 1/2)(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.

Revised on January 31, 2012 18:22:33 by Todd Trimble (74.88.146.52)