Non-simple groups $G$ with only non-trivial quotient isomorphic to $G$

Question

If $G$ is a group such that every non-trivial subgroup is isomorphic to $G$ itself, then $G= \mathbb{Z}$ is the only infinite group with that property (up to isomorphism). Amongst the finite groups we essentially have $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime.

Turning arrows around, let us say that a group $G$ has property $\text{Q}$ if

whenever $H$ is a non-trivial group and $s:G\to H$ is a surjective group homomorphism, then $H\cong G$.

Again $\mathbb{Z}/p\mathbb{Z}$ has this property for $p$ prime. (I don't know whether they are the only finite groups with property $\text{Q}$.) Note that $\mathbb{Z}$ does not have property $\text{Q}$.

Question. For which cardinals $\kappa\geq\aleph_0$, if any, is there a non-simple group with property $\text{Q}$ having cardinality $\kappa$?

Acknowledgement. Thanks to Dave Benson for suggesting to consider non-simple groups only.

Answer 1

CW answer.

As already mentioned in a comment, the question was already asked at MathSE. The property in consideration is "every nontrivial quotient is isomorphic to the group itself".

In summary:

• the trivial group satisfies this property
• simple groups satisfy this property
• as mentioned in this answer, finitely generated groups with this property are trivial or simple (indeed, every nontrivial f.g. group has a simple quotient). This more generally holds for groups that are finitely normally generated.
• as mentioned in the same answer the most obvious nontrivial nonsimple examples are Prüfer (aka quasi-cyclic) groups $C_{p^\infty}$ for $p$ prime. These are the only ones among abelian groups.
• nonabelian nonsimple examples are provided in this answer. (As the Prüfer groups, they are locally finite; these ones are direct limits of iterated wreath products of finite alternating groups.)

Answer 2

References & terminology.

An algebraic structure is called pseudosimple if it has more than one element and is isomorphic to all its nontrivial quotients. This concept was introduced in the paper

Monk, Donald
On pseudo-simple universal algebras.
Proc. Amer. Math. Soc. 13 (1962), 543-546.

The main result of the paper is that if $A$ is pseudosimple algebra, then the congruence lattice of $A$ is a well-ordered chain of order type $\omega^{\beta}+1$ for some ordinal $\beta$, and, conversely, for every ordinal $\beta$ there exists a pseudosimple lattice whose congruence lattice is a well-ordered chain of order type $\omega^{\beta}+1$.

A later paper that is closer to the problem on this page is:

Schein, Boris M.
Pseudosimple commutative semigroups.
Monatsh. Math. 91 (1981), no. 1, 77-78.

The main result here is that a commutative semigroup is pseudosimple iff it is a $2$-element commutative semigroup, $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$ for some prime $p$. This paper states as a corollary to the main result that an abelian group is pseudosimple iff it is isomorphic to $\mathbb Z_p$, or $\mathbb Z_{p^{\infty}}$