Questions tagged [hopfian-groups]

For questions about the Hopf property in groups and its failure.

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Groups with a dense chain of (isomorphic) subgroups

Are there any interesting non-Abelian examples of groups which are equal to the union of a dense chain of normal subgroups, with each subgroup isomorphic to the original group. That is, a group $G$ ...
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The Hopfian property for groups

Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context ...
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When can a pair of groups be embedded in each other?

This is a question I made up, but couldn't solve even after some days' thought. Also if any terminology is unclear or nonstandard, please complain. Given groups $G$ and $H$, we say that $G$ can be ...
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$\langle X|\emptyset\rangle\ncong\langle X|R\rangle$ for finitely presented groups (exercise in Massey)

Let $:F_n$ denote the free group of rank $n$. How can I solve the exercise 7.6.3.(b), page 234, in Massey's Algebraic Topology? I'm guessing there's been made a mistake and (b) actually reads "If $G$ ...
Is there an epimorphism $f\colon \mathrm{GL}(2,\mathbb{Z})\to \mathrm{GL}(2,\mathbb{Z})$ which is not injective? Here, $\mathrm{GL}(2,\mathbb{Z})$ is the group of invertible $2\times 2$ matrices with ...
The free group $F_3$ being a quotient of $F_2$
Every finitely generated free group is a subgroup of $F_2$, the free group on two generators. This is an elementary fact, as is the fact that $G$, finitely presented, is the quotient of $F(|S|)$ the ...