Let $G$ be finite group with minimal number of generators$d$, and all his proper subgroups have at most $d-1$ as minimal number of generators. Fix a normal subgroup $N$ of $G$. For all subgroups $H$ of $G$ satisfying $[HN,N] \subseteq N^2$; consider the transfer homomorphism $HN \rightarrow H/H^2$. Is it surjective?
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$\begingroup$ What do $N^2$ and $H^2$ mean? $\endgroup$– Derek HoltCommented Apr 6, 2021 at 22:04
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$\begingroup$ $N^2$ is the subgroup generated by all the squares of $N$, the same thing for $H^2$. $\endgroup$– A.MessabCommented Apr 6, 2021 at 22:14
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4$\begingroup$ Suppose we let $G = C_2 \times C_2$ be a Klein $4$-group and $H$ and $N$ distinct subgroups of order $2$. Is that not a counterexample? The transfer map is $g \mapsto g^2$, so the image is trivial. $\endgroup$– Derek HoltCommented Apr 6, 2021 at 22:33
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$\begingroup$ @DerekHolt thank you very much professor, it is indeed a counterexample. $\endgroup$– A.MessabCommented Apr 8, 2021 at 9:50
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