There are many statements in abstract algebra, often asked by beginners, which are just *too good to be true*. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup of $G$? As an experienced mathematician, we see immediately that *there is no reason for this to be true* — even without thinking about this in detail. Often we can quickly come up with counterexamples. Sometimes, it is hard to find counterexamples.

Many questions fall into this category, for example:

- If $f : R \to S$ is a ring homomorphism and $I \subseteq R $ is an ideal, is then $f(I) \subseteq S$ an ideal? (SE/2200335)
- $\DeclareMathOperator\Aut{Aut}$If $G,H$ are groups, do we have $\Aut(G \times H) \cong \Aut(G) \times \Aut(H)$? (SE/1236571)
- Is every submodule of a finitely generated module also finitely generated? (SE/83078)
- If $A$ is an abelian group with $A^3 \cong A$, does this imply $A^2 \cong A$? (MO/10128)
- If $A$ is an abelian group with $A \oplus \mathbb{Z}^2 \cong A$, does this imply $A \oplus \mathbb{Z} \cong A$? (MO/218113)
- If $G$, $H$ are groups whose group algebras $ \mathbb{Q}[G]$, $\mathbb{Q}[H]$ are isomorphic, are then $G$, $H$ isomorphic? (SE/1342851)
- see also MO/23478 for common false beliefs in mathematics

But my question is actually about situations where, for some strange reason, our first gut feeling is not correct and a **wrong-looking statement turns out to be true**. Examples will be abundant, which is why I want to restrict this question to examples coming from abstract algebra (you are welcome to open similar questions for other branches and flavors of mathematics, and please let me know if there are already questions of this type).

Here are some examples which come to my mind:

- Every group homomorphism $\mathbb{Z}^{\mathbb{N}} \to \mathbb{Z}$ is a finite linear combination of projections. In fact, $(\mathbb{Z}^{ \mathbb{N}})^* \cong \mathbb{Z}^{\oplus \mathbb{N}}$. (Specker 1950)
- If $A$, $B$ are
*finitely generated*abelian groups (more generally, finitely generated modules over a commutative Noetherian ring) and $f : A \to A \oplus B$, $g : A \oplus B \to B$ are homomorphisms such that $0 \to A \xrightarrow{f} A \oplus B \xrightarrow{g} B \to 0$ is exact, then it is split exact. - If $A$, $B$, $C$ are
*finite*groups such that $A \times B \cong A \times C$, then $B \cong C$. (SE/3579745) - every negation of the examples mentioned above, for example: There
*is*an abelian group $A$ with $A \cong A^3$ and $A \not\cong A^2$. (However, I am more interested in "positive" results.)

**I am looking for statements in abstract algebra where this is your reaction when you learn that they are actually true.**

Please try to include a reference for the statement and proof.

15more comments