i have a quick doubt, if i have $G$ a group and say, $H,K$ subgroups of $G$ and we know that $HK\trianglelefteq G$ can we conclude that $H$ and $K$ are normal in $G$? i was thinking of letting $H$ be the neutral element, but im not sure if it will work.
2026-03-31 14:23:20.1774967000
Normality subgroups of a normal group
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No. Let $G = S_3$ and $H = \langle (1 \; 2)\rangle$, $K = \langle (1 \; 2 \; 3)\rangle$. Then $HK = G$ is a normal subgroup, but $H$ is not a normal subgroup.