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It is important to understand the relationship of normal subgroups to their parent. One concept that needs to be understood is whether the normality of a subgroup does not depend on which parent group it is a subgroup of. This is true for some subgroups ($\{e\}$ is always normal in any group) but is it generally speaking true for every group? The question can be phrased in symbolic form: let $G, K$ be groups and $H \leq G$ and $H \leq K$. If $H \lhd G$ then necessarily $H \lhd K$?

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    $\begingroup$ This question shouldn't be closed because this question is about "understanding mathematical concepts and theorems" and so is within the scope of this website. $\endgroup$ – BalancedTryteOperators Feb 9 at 20:07
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    $\begingroup$ then you should add some context. As it stands it's a Problem Statement Question (PSQ), which the site tends to dislike. Since it's about "understanding mathematical concepts and theorems" perhaps you could clarify which ones exactly, and motivate the question? $\endgroup$ – postmortes Feb 9 at 21:30
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    $\begingroup$ This is a sufficiently natural question to require no motivation. $\endgroup$ – darij grinberg Feb 9 at 22:28
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    $\begingroup$ @darijgrinberg With reference to the original version of this question (and not the current version, which is much improved), I don't really believe that there is such a thing as a question that is so "natural" that it requires no motivation. At the very least, one should give enough context to explain why it is a natural question. $\endgroup$ – Xander Henderson Feb 9 at 23:47
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    $\begingroup$ @XanderHenderson: It suffices to read a text which often talks about "normal subgroups" without specifying the parent group to ask this question. $\endgroup$ – darij grinberg Feb 9 at 23:48
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Normality is relative to the parent group. This is evident in the way it is defined. For a group $G$ and subgroup $N$ of $G$ we say that $N$ is normal in $G$ if for all $g \in G$ we have $N^g=N$. What brings the parent group in the definition is the "if for all $g \in G$" bit.

To give a concrete example: every group is a normal subgroup of itself, clearly. Now let $G$ be a non-abelian finite simple group and let $H$ be a proper and non-trivial subgroup of $G$. Then $H$ is normal in itself, but it is not normal in $G$.

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Yes.

$S_4$ has a Sylow-3 subgroup $\mathbb{Z}_3$, but it is not normal because the number of Sylow-3 subgroups $> 1$. And a Sylow-p subgroup is normal iff. $n_p = 1$. $S_3$ has $\mathbb{Z}_3$ as a normal Sylow-p subgroup for the same reason.

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