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I know from here that any normal subgroup $K$ of a normal subgroup $H$ in a group $G$ is not necessarily normal in $G$. But I was wondering if this is true in the following case:

Let $G$ be a group. Let $H$ be a normal subgroup of the commutator subgroup $(G,G)$. Is $H$ a normal subgroup of $G$ ?

Thanks in advance for your enlightenment?

K. Y.

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1 Answer 1

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Consider $G=A_4$, the alternating group of order $12$. Its derived subgroup is $V_4$, the unique Sylow $2$-subgroup of $A_4$. As $V_4$ is Abelian, any of its $2$-element subgroups $H$ is normal in $V_4$, say $H=\left<(1\,2)(3\,4)\right>$. But $H$ is not normal in $A_4$.

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  • $\begingroup$ Thanks for your answer. Is there any condition we can add for the statement to be true? $\endgroup$
    – Kal_Aki
    Commented Jun 3, 2018 at 14:20
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    $\begingroup$ If we take $H$ to be a characteristic subgroup of $[G,G]$, then $H$ will be normal in $G$. $\endgroup$
    – Skyhit2
    Commented Jun 3, 2018 at 14:23
  • $\begingroup$ Or $H$ a characteristic subgroup of any normal subgroup of $G$, again $H$ is normal in $G$. $\endgroup$
    – Leppala
    Commented Jun 4, 2018 at 9:31

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