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A question regarding a similar this has already been asked and answered: If $G$ is a nilpotent group and $H\leq G$ with $H[G,G]=G$ then $H=G$.

However, what I am asking for a reference of a slight generalization.

Let $G$ is nilpotent, and $H,N$ subgroups of $G$, with $N$ normal. If $N[H,G]=H$, then $N=H$.

The proof for this is very similar to the less general case: if $H[G,G]=G$, then $H=G$.

Does anyone know where I can find a reference for this, let's call it, Lemma?

Thanks in advance.

Gillyweeds
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    How about $G=\langle a,b \mid a^4=b^2=(ab)^2=1\rangle = D_8$, $H=\langle a^2,b \rangle$, $K=\langle b \rangle$? – Derek Holt Sep 09 '24 at 11:00
  • You are right, I think I we need $K$ to be $G$-invariant, i.e. normal. – Gillyweeds Sep 09 '24 at 14:01
  • In that case your should edit the question. – Derek Holt Sep 09 '24 at 15:49
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    Moding out by $N$, you are trying to show that $[H,G]=H$ implies $H$ trivial. – Arturo Magidin Sep 09 '24 at 16:09
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    IMO the most likely reference will be found after applying Arturo's simplification. Then the result you need is just $[H,_i G]\le\gamma_{i+1}(G)$, which can be found many books (personally I'd try Hall's nilpotent group notes). But also this is trivial to prove directly. – Steve D Sep 09 '24 at 18:36
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    This is Exercise 1.5(i) in Susan McKay's Finite $p$-groups, Queen Mary Maths Notes 18., p. 9 – Arturo Magidin Sep 10 '24 at 14:43

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