# Is the normality of a subgroup dependent on which group is its parent?

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 $$H \lhd K$$?

• 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. – BalancedTryteOperators Feb 9 at 20:07
• 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? – postmortes Feb 9 at 21:30
• This is a sufficiently natural question to require no motivation. – darij grinberg Feb 9 at 22:28
• @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. – Xander Henderson Feb 9 at 23:47
• @XanderHenderson: It suffices to read a text which often talks about "normal subgroups" without specifying the parent group to ask this question. – darij grinberg Feb 9 at 23:48

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 $$G$$.
$$S_4$$ 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.