In Paul Cohn's Universal Algebra book, p. 61, he writes
"[…]Every $\Omega$-algebra $A$ has itself and the trivial algebra as homomorphic images. If it has no others and is non-trivial it is said to be simple. By [the first isomorphism theorem] an $\Omega$-algebra $A$ is simple if and only if has precisely two congruences, namely $A^2$ and $\Delta$ [the diagonal]."
Here, as usual, we identify algebras up to isomorphism.
I am pretty sure this is wrong. At least, the argument does not follow, as a quotient of a strucutre $A$ by a non-trivial congruence may be still isomorphic to $A$ (e.g. the multiplicative circle $\mathbb{S}^1\subseteq\mathbb{C}\setminus\left\{0\right\}$ modulo $\left\{\pm 1\right\}$, as a structure in the signature of groups; or any infinite set modulo a relation that identifies two points, in the empty signature.)
So I was trying to find an example of a non-simple structure $A$ (in the sense that it has nontrivial congruences) for which all non-trivial homomorphic images are isomorphic to $A$. I'd suppose this is possible even for groups, but could not find a concrete example
Question: Is there a non-simple group $G$ whose homomorphic images are all isomorphic to either $G$ or to the singleton group $\{1\}$?
I thought a bit of $G=\mathbb{Q}/\mathbb{Z}$ but got nowhere.
Best Answer
An algebraic structure that is isomorphic to each of its nontrivial quotients is called pseudosimple. This concept was introduced in:
Monk, Donald
On pseudo-simple universal algebras.
Proc. Amer. Math. Soc. 13 (1962), 543-546.
The pseudosimple commutative groups are known, they are the Prüfer groups $\mathbb Z_{p^{\infty}}$. The pseudosimple commutative semigroups are known to be either simple semigroups or else one of the Prüfer groups considered as a semigroup.