Counterexample bank

False — $A_{4}$ has no subgroup of order 6

False — $\mathbb{Z}[\surd-5]: 6 = 2\cdot3 = (1+\surd-5)(1-\surd-5)$

False — f(z) = sin(z)/z at 0

False — $A_{4}$ has order 12 but no subgroup of order 6

False — ⟨(12)(34)⟩ ⊴ $V_{4}$ ⊴ $A_{4}$

False — $A_{4}$ has the normal subgroup $V_{4} =$ {e, (12)(34), (13)(24), (14)(23)}

False — Impossible — if G/Z is cyclic then G is abelian, so G/Z is trivial

False — $S_{3}$ of order 6 = 2·3

False — (X) in $\mathbb{Z}[X]$, or (0) in any integral domain that is not a field

False — 2 in $\mathbb{Z}[\surd-5]$

False — $x^{4} + 1$

False — The field of all algebraic numbers Q̄ over $\mathbb{Q}$

False — $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ has degree 3 but only the identity automorphism

False — $A = \mathbb{Q}, B = \mathbb{R}$∖$\mathbb{Q}$ in $\mathbb{R}$

False — id : $(\mathbb{R}$, discrete$) \to (\mathbb{R}$, usual), or t ↦ (cos t, sin t) from $[0, 2\pi)$ to $S^{1}$

False — The unit ball of $\ell^{2}$

False — $\mathbb{Q}$ with the usual topology

False — An infinite set with the cofinite topology

False — $f(z) = ({\bar{z}})^{2}/z$ for z ≠ 0, f(0) = 0

False — sin(1/z) on $\mathbb{C}$∖{0}

False — f(z) = 1/z on the annulus ½ < |z| < 2

False — f(z) = eᶻ on {Re z < 0}

False — f(z) = sin z / z at 0

False — The indentation at 0 when computing $\int_{0}^\infty \sin x/x$ dx

False — f(z) = eᶻ on $\mathbb{C}$

False — Liouville's theorem