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What is a normal subgroup? I have heard all of the following:

  1. A subgroup $N$ of $G$ is normal if $gNg^{-1} = N$ for all $g \in G$. Or equivalently, a subgroup is normal if it is invariant under conjugation by elements of $G$, that is, if $gng^{-1} \in N$ for all $n \in N$ and $g \in G$.

  2. A subgroup is normal if it is preserved under "change of vantage point" [1]. I have also seen it described as: A subgroup is normal when it is "relativistically invariant" [2].

  3. Normal subgroups are those subgroups that are kernels of homomorphisms.

  4. Normal subgroups are those subgroups that you can create a quotient group with.

  5. A subgroup is normal if every left coset equals the corresponding right coset.

Now, this is all very interesting and I have read proofs of why some of these statements are equivalent, but I still don't have the "big picture" and "intuition" of why all these statements are basically describing the same thing. [Feel free to correct me if I've misunderstood/mischaracterized some of the statements or equivalences.] I look at these statements and I think, "that's all really cool, but can I (quickly) see why?"

I am hoping for anyone to provide some clear, intuitive, and simple reasoning (and preferably also some "big picture"/high-level overview) as to why each statement is essentially equivalent to the others.

For example, why should "invariant under conjugation" be equivalent to "kernel of a homomorphism"? What does "preserved under change of vantage point" or "relativistically invariant" mean specifically, and what, say, does this have to do with the formation of quotient groups?

Sources:

[1] https://math.ucr.edu/home/baez/normal.html

[2] See Per Vognsen's answer in https://mathoverflow.net/questions/38639/thinking-and-explaining, in quoting V.I. Arnold.

Shaun
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twosigma
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    By definition of a two-sided identity element, $eg=ge=g$ for all group elements. A normal subgroup is in some sense the identity element $,e,$ "puffed up" to form a subgroup which satisfies the analogous property. For example, property $5$ for cosets. – Somos Mar 04 '21 at 02:51
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    Given a homomorphism of algebras (in the sense of universal algebra) $f\colon A\to B$, $f$ induces a natural equivalence relation on $A$ by $x\sim_f y \iff f(x)=f(y)$. You can then define an algebra structure on $A/\sim_f$ in the “obvious way”. If you do this with groups, the equivalence relations are completely determined by the equivalence class of $e$, because $x\sim_f y \iff (x)=f(y)\iff f(x)f(y)^{-1}=e\iff f(xy^{-1})=f(e)\iff xy^{-1}\sim_f e$. This is a subgroup (because $x\sim_f y$ and $z\sim_f w$ imply $xz\sim_f yw$); the subgroups that arise this way are exactly the normal ones. – Arturo Magidin Mar 04 '21 at 03:36
  • I don't think there's any better way to see these things than to study the proofs, which are in every introduction to group theory (and probably in many earlier questions on this website). Well, except for the "change of vantage point" and "relativistically invariant" ones, I'd never seen those before. – Gerry Myerson Mar 04 '21 at 03:49
  • My favorite interpretation of normal subgroups is explained in https://math.stackexchange.com/a/1014784/589 – lhf Mar 04 '21 at 10:04

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