4

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.

Luiz Cordeiro
  • 18,513
  • 1
    You are on the right track. Try the $p$-primary part of ${\mathbb Q}/{\mathbb Z}$ for a fixed prime $p$, also known as the Prüfer $p$-group – Derek Holt Jun 22 '22 at 19:50
  • 1
    I would not say the book is wrong, so much as not very well-formulated. I think it's fairly clear what it's trying to say, although you are right that technically it's a mistake. – Captain Lama Jun 22 '22 at 20:01
  • Related question: https://math.stackexchange.com/questions/768561/sort-of-simple-non-hopfian-groups/768729#768729 – Jeremy Rickard Jun 22 '22 at 22:22

1 Answers1

7

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.

Keith Kearnes
  • 13,798
  • Very nice! Is anything known about non-commutative pseudosimple groups? – Vincent Jun 22 '22 at 20:06
  • 2
    @Vincent: I don't know that anyone has considered this for noncommutative groups. There is a general result which implies the following for groups: if $G$ is a pseudosimple group that is not simple, then $G$ is not finitely generated. [Nonfinite generation holds for any nonsimple pseudosimple algebraic structure of finite signature.] – Keith Kearnes Jun 22 '22 at 20:39