This is a question about Hirshon's paper "The center and the commutator subgroup in hopfian groups"(https://projecteuclid.org/euclid.afm/1485894451). Theorem 12 stating that if $B$ is a perfect group with finitely many normal subgroups and $A$ is a hopfian group, then the direct product $A \times B$ is hopfian. Is it possible to weaken the "finitely many normal subgroups" hypothesis?
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$\begingroup$ This is a weird question... are you asking whether every perfect group has finitely any normal subgroups? or whether Hirshon skipped a hypothesis (in which case you should elaborate) $\endgroup$– YCorCommented Dec 5, 2017 at 9:14
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$\begingroup$ projecteuclid.org/euclid.afm/1485894451 $\endgroup$– YCorCommented Dec 5, 2017 at 9:15
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$\begingroup$ It's impossible to understand your question without looking at the paper; I just did. It's indeed not very clear what are the assumptions: the statement of the theorem is "If $B$ is a perfect group then $A\times B$ is hopfian.". The abstract summarizes the results. It seems that the only standing assumptions are that $A$ and $B$ are hopfian (and, in this precise theorem, that $B$ is perfect). $\endgroup$– YCorCommented Dec 5, 2017 at 9:22
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$\begingroup$ @Ycor Thanks for providing the link. I think $B$ doesn't have to be assumed to be hopfian. But the hidden hypothesis "finitely many normal subgroups" is needed in Hirshon's proof. $\endgroup$– Shijie GuCommented Dec 5, 2017 at 19:24
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1$\begingroup$ Certainly this "finitely many normal subgroups" is an overkill and should be weakened. The most optimistic statement would be that $A,B$ hopfian implies $A\times B$ hopfian (this would be an if-and-only-if). Unfortunately it's not mentioned, and I can't find any reference in Hirshon's survey "sciencedirect.com/science/article/pii/…" (where he addresses as open problem whether hopfian is stable under taking direct product with $\mathbf{Z}$) $\endgroup$– YCorCommented Dec 5, 2017 at 23:18
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