If I have a quotient group $H/H_0$ and another group $G$ such that $H_0G\unlhd HG$. Is it then true that $H/H_0 = HG/H_0G$?
2026-04-06 05:54:30.1775454870
Is $H/H_0 = HG/H_0G$?
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No, take $\{1\}=H_{0} \lt H \lt G$, and put $G$ in your example the whole group $G$. Then you get a contradiction: $H/H_{0}=H=G/G\cong \{1\}$
But ... it is true when for instance $G \cap H\subseteq H_{0}$. Let us prove this. It hinges on the fact that in general $H \cap H_{0}G=H_{0}(H \cap G)$ (Dedekind's Modular Law). Then $HG/H_{0}G=H(H_{0}G)/H_{0}G\cong H/(H \cap H_{0}G)=H/H_{0}(H \cap G)=H/H_{0}$, if $G \cap H\subseteq H_{0}$.