calculating core in a quotient group Let $G$ be a finite group with $N\trianglelefteq G$ and $H\le G$. I would like to know if $core_{G/N}(HN)=core_G(H)N$. My 'proof' is that $core_{G/N}(HN)=\cap_{g\in G}NgHNNg^{-1}=\cap_{g\in G}gHg^{-1}N=core_G(H)N$, but I'm finding in some (long-winded so I won't include them here) calculations that this doesn't seem to hold. Obviously my calculations could be wrong, but to save me searching through the is there a problem in the above proof?

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ewcommand{core}{\mathrm{core}}$I think the correct way of writing it is $\core_{G/N}(HN/N) = \core_{G}(H) N / N$.

But it does not hold true anyway.

Think of $G = S_{3}$, $N = \Span{(1 2 3)}$, $H = \Span{(1 2)}$.

We have $\core_{G}(H) = \Set{1}$, but $H N /N = G / N$, so $$\core_{G/N}(H N/ N) = G / N \
e \core_{G}(H) N / N = N / N.$$

The point being that in general $$ \bigcap_{g\in G} (gHg^{-1}N) \supsetneq (\bigcap_{g\in G} gHg^{-1}) N. $$