Talk:Normal subgroup

Truly terrible writing
It is beyond idiotic to write, as in the Definition section:

"''A subgroup $N$ of a group $G$ is called a normal subgroup if it is invariant under conjugation; that is, the conjugation of an element of $N$ by an element of $G$ is always in $N$:
 * $$N \triangleleft G\, \Leftrightarrow\; \forall\; n\in N,\; \forall\ g\in G\colon\; gng^{-1}\in N.$$''"

where the "normal subgroup symbol" (the horizontal triangle) is used to mean "is a normal subgroup of" in the article Normal subgroup ... despite the fact that this notation has not been introduced in this article (at least not before it is used here).

You might as well be writing in Aramaic. Do you not care whether what you write is understood? Maybe it would be better if you left editing Wikipedia mathematics to people who comprehend how to communicate.2600:1700:E1C0:F340:1D3D:4D5C:F18F:F5CA (talk) 07:01, 13 November 2018 (UTC)


 * WP:SOFIXIT. --JBL (talk) 22:22, 2 December 2018 (UTC)

It would really be great to start these math articles off with intuition rather than definition. The definition, while accurate is only useful to someone who already knows what the heck this is and us trying to remember the definition! Bengom (talk) 15:46, 27 August 2019 (UTC)

More intuitive intro?
The current intro is not useful unless you are already familiar with the topic.... Bengom (talk) 15:48, 27 August 2019 (UTC)
 * No, I found it useful for learning the definition of a normal subgroup when I had just learned the basic definition of a group. If you have specific suggestions on how to make it more intuitive, then feel free to place them here. However, one cannot really talk about normal subgroups without knowing what a group is in the first place. More intuitive notions such as "the left and right cosets coincide" and "the parent group acts on the normal subgroup by conjugation" require more definitions than the basic definition of a group.--Jasper Deng (talk) 16:14, 27 August 2019 (UTC)
 * The definition that is most informative to me is that normal subgroups are preserved under inner automorphisms. Essentially that requires understanding that groups are symmetries, and that these symmetries act on the group itself, and so in a sense the normal subgroups are symmetric inside the group. Although typically people learn about normal subgroups before learning about group actions and automorphisms, so I don't know if it makes sense to explain that in the introduction. Woscafrench (talk) 16:29, 28 August 2019 (UTC)

Routine proofs of basic facts?
Recently, added (diff) some routine proofs of basic properties, namely, that the product in the quotient G/N is well defined when N is normal. This strikes me as extremely textbooky (in the sense of WP:NOT) -- using this level of detail for this fact is consistent with writing a textbook chapter on normal subgroups, and not consistent with writing an encyclopedia article (where essential facts can be highlighted without the burden of being carefully proved, thus making it easy for a reader to find the key information quickly). I reverted, but LI re-reverted; rather than start an edit war, I'd like to request input from other editors here. --JBL (talk) 11:38, 7 October 2019 (UTC)


 * Indeed, I don't mean to start an edit war, so let's wait for someone else to give their opinion. I just want to say here that many other pages include elementary proofs of basic facts. The first related examples that come to mind are in the Group page. See e.g. here where the arguably elementary and "textbooky" proof of the uniqueness of inverses is given. There are countless other examples of this kind. The page on normal subgroups, on the other hand, is rather devoid of any detail and proofs. I appreciate the usefulness of having a nice and tidy list of results, but one can have it both ways: tidy list of results with proofs attached. For example by having the proofs in collapsible headings (as done e.g. in this page). Luca (talk) 12:02, 7 October 2019 (UTC)

Normal subgroup example
I think the example subgroup has a small error.

It says that in S_3, (123)H = {(123),(13)} and H(123) = {(123),(23)}. I think the left and right multiplication have been reversed here: (123)H = {(123)(1), (123)(12)}, (123)(12) is equal to (23), not (13). It's been a long time since I took my group theory class and I was mostly using this article to brush up, but I just wanted to confirm whether or not this was a mistake. Sorry if I'm the one who's made the error. — Preceding unsigned comment added by 2603:6011:A546:C100:C4DC:69FF:D634:8130 (talk) 22:18, 11 May 2021 (UTC)
 * There are two opposite conventions for how to multiply permutations. Under the convention being used in the article, (123) is the function that sends 1 to 2, 2 to 3, and 3 to 1, (12) is the function that sends 1 to 2 and 2 to 1 and 3 to 3, and multiplication is composition of functions from right to left.  With this convention, (123)(12) first applies (12), then applies (123), so it sends 1 to 3 (first (12) sends 1 to 2, then (123) sends that to 3), 2 to 2, and 3 to 1, or in other words (123)(12) = (13).  This convention is totally fine, and has the virtue that it agrees with the usual order in which functional composition is written.  The reverse convention (where we apply the permutations in a product starting with the leftmost first) is also totally fine, and has the virtue that it agrees with the usual order in which one reads the English language.  It is only when two people using different conventions (as, for example, you and the person who wrote the example) try to communicate with each other that there is an issue. --JBL (talk) 22:58, 11 May 2021 (UTC)