Does $G\cong G/H$ imply that $H$ is trivial?

Question

Let $G$ be any group such that

$$G\cong G/H$$ where $H$ is a normal subgroup of $G$.

If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In what kind of group could I search for a counterexample ?

Answer 1

Look at $G=\bigoplus\limits_{i=1}^\infty\ \mathbb Z$ and the subgroup $H=\mathbb Z\ \oplus\ \bigoplus\limits_{i=2}^\infty\ 0$.

Answer 2

For a finitely generated non-abelian example, see the Baumslag-Solitar groups, in particular $\mathrm{BS}(2,3)\cong \langle a,b; b^{-1}a^2b=b^3\rangle$.

Answer 3

Sticking with an abelian theme, you could let $G=\mathbb{Z}(p^{\infty})$ and $H=\mathbb{Z}(p^{k})$ for a positive integer $k$.

Answer 4

If $G\cong G/H$ implies $H$ is trivial then $G$ is called Hopfian. Otherwise, $G$ is called (imaginatively!) non-Hopfian. Non-Hopfian-ness is a truly nasty property!

A related property is that of Residual finiteness. A group is residually finite if for any two elements $g, h\in G$ there exists some homomorphism, $\phi$, to a finite group $H$, $\phi:G\rightarrow H$ such that $\phi(g)\mathrel{\neq_H} \phi(h)$. Equivalently, noting that $\phi(gh^{-1})=1$, $G$ is Residually finite if for any $g\in G$ there exists some $\phi:G\rightarrow H$, $H$ finite, such that $\phi(g)\mathrel{\neq_H} 1$. Note that a finite group is clearly residually finite, and the proof that a finite group is Hopfian is low-level.

It is interesting to note that if $G$ is finitely generated and non-Hopfian then $G$ is not Residually finite. A proof of this can be found in many/most graduate-level texts which cover infinite groups (for example, D. J. S. Robinson's book A Course in the Theory of Groups), or see this Math.SE answer. As has been pointed out in the comments below, this only holds if $G$ is finitely generated. For example, the free group on countably many generators is residually finite but is non-Hopfian.

As Joseph Cooper pointed out in his answer, certain Baumslag-Solitar groups are non-Hopfian, such as $$BS(2, 3)\cong \langle a, b;b^{-1}a^2b=a^3\rangle$$ (to see this, take the map $a\mapsto a^2, b\mapsto b$ and play around with it for a bit - it has non-trivial kernel, but the proof is non-trivial...you can find a proof in Magnus, Karrass and Solitar's book Combinatorial Group Theory, in their section on one-relator groups, or in this Math.SE answer). Indeed, there exists a classification of the Baumslag-Solitar groups with respect to their Hopficity and Residual-finiteness. Specifically,

The group $BS(m, n)=\langle a, b; b^{-1}a^mba^{n}\rangle$ for $m, n\in\mathbb{Z}$ is,

• Residually finite if and only if $|m|=|n|$ or $|m|=1$ or $|n|=1$,

• Hopfian if and only if it is Residually finite or the set of prime divisors of $m$ is equal to the set of prime divisors of $n$.

So, for example, taking $m$ and $n$ to be coprime and each of absolute value greater than $1$ will yield a non-Hopfian group. There is a paper of Meskin which generalises this to groups with with relation where the relation is of the form $uv^mu^{-1}v^n$ where $u, v$ are words in the generators. Indeed, his statement about residual-finiteness is identical to the case of the Baumslag-Solitar groups ($G$ is residually finite if and only if $m$ and $n$ are equal to each other or one in absolute value). Edit: I believe that Meskin's result on residual finiteness was the original result. That is, he proved the general case where the relation has the form $uv^mu^{-1}v^n$ and as a corollary got the first proof of the Baumslag-Solitar case.

Also, the fact that some Baumslag-Solitar groups are not Hopfian is surprising, as the Baumslag-Solitar groups are the $HNN$-extensions of $\mathbb{Z}$, and one would think that an extension of $\mathbb{Z}$ would be very nice indeed! However, that is clearly not the case...

(Note that Chris Leary asked only a few days ago what the history of Hopfian groups was - see here.)

Answer 5

A different kind of example giving a kind of realization of the Frobenius kernels. Let $p$ be a prime, $F=F_p=GF(p)$, let $x$ be an unknown, and let $R$ be the set of finite $F$-linear combinations of $x^q$, where $q$ is a non-negative rational number with the property that $p^\ell\cdot q$ is an integer for some natural number $\ell$. Then $R$ is a ring w.r.t. the natural operations. Let $A=R/I$ where $I$ is the ideal of $R$ generated by $x^p$. Then $\phi:A\rightarrow A, a\mapsto a^p,$ is a surjective endomorphism of $F$-algebras with kernel $J=$ the ideal generated by the coset $x+I$. Let $n\ge1$ be an integer. By functoriality of the affine group scheme $GL_n$ the mapping $\phi$ gives (acting entrywise) a surjective homomorphism of groups $\phi:GL_n(A)\rightarrow GL_n(A)$. So, with $G=GL_n(A)$ and $H=\ker\phi$, we get the desired isomorphism $G/H\simeq G$. The subgroup $H$ consists of those matrices that are congruent to the identity matrix modulo $J$.