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Dear All! I guess this should be well-known, but I cannot find it in literature:

Is it true that if $H$ is a subgroup of finite index in $G$, and $H$ is hopfian, then $G$ is also hopfian?

In case when the groups are finitely generated, the answer is "yes" by R. Hirshon.

(And the other way around question has answer no, as was kindly answered by Kate YuschenskoJuschenko, even for finitely generated case -- which is quite bizarre!!!)

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Dear All! I guess this should be well-known, but I cannot find it in literature:

Is it true that if $H$ is a subgroup of finite index in $G$, and $H$ is hopfian, then $G$ is also hopfian?

In case when the groups are finitely generated, the answer is "yes" by R. Hirshon.

(And the other way around question has answer no, as was kindly answered by Kate Yuschensko, even for finitely generated case -- which is quite bizarre!!!)

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Dear All! I guess this should be well-known, but I cannot find it in literature:

Is it true that hopficity passes to subgroups/supergroups of finite index?

If not, what happens if we additionally require finite generation for the groups?

Right, and what I should mention: R. Hirshon proved that hopficity passes from $H$ is a subgroup of finite index to a groupin $G$, and $H$ is hopfian, if they then $G$ is also hopfian?

In case when the groups are finitely generated, the answer is "yes" by R. Hirshon.

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