Talk:Isomorphism theorems

The first theorem

Following herstein, I'm not familiar with the concept of the "image of a mapping" as used in the statement of the first theorem. a mapping has a range (or codomain), and any subset of it's domain has an image under the mapping. But the "image of a mapping" is a concept I've not encountered.

If I interpret "image of a mapping" to mean it's codomain, then the 3rd list item is restating the conclusion of the "in particular" qualification that immediately follows, only it omits the requirement for the mapping to be surjective -- which makes it wrong. — Preceding unsigned comment added by 5.102.207.31 (talk) 03:03, 7 April 2014 (UTC)

The image of the map is the image of the domain under the map. Magidin (talk) 16:42, 7 April 2014 (UTC)

Noether's Isomorphism Theorems

Editor Sbilley made edits adding Noether's name to most references to the isomorphism theorems. The edit summary says that the editor "was tasked" with updating the name; tasked by whom, exactly? Has this been discussed and consensus reached? Further, the proffered justification for the changes is that in Math "it is typical to attribute theorems to the original author". Things are called what they are called, not what they ought to be called (Quadratic Reciprocity is called "Quadratic Reciprocity", not "Gauss's Law of Quadratic Reciprocity", for example). I am not familiar with any standard reference to the isomorphism theorems that calls them "Noether's Isomorphism Theorems". Even if the attribution is accurate, if the theorems are not called that in the literature, then they should not be called that in the article; an attribution is certainly in order, or even a brief mention in the lede, but it seems to me that the kind of wholesale re-branding that this editor is doing here is ill-advised at best, and in any case needs consensus. Magidin (talk) 22:08, 27 October 2016 (UTC)

This naming convention has a long sorted history. Karen_Smith_(mathematician) gave the Noether lecture at the Joint Meetings of the American Mathematical Society and the Mathematical Association of America's in January of 2016 about covering the history. Her argument was very well received that we should be using the name "Noether's Isomorphism Theorems" to follow the standard naming conventions for important contributions in math. She also discussed several sources that have been using this name going back to the time shortly after Noether's paper was originally published. Here are some sources that refer to these theorems as Noether's Isomorphism Theorems:

(1) "Graduate Algebra: Commutative View" by Louis Rowen.

(2) "Commutative Algebra" by A. Altman

(3) "Algebraic Topology" by Edwin Spanier

(4) "Algebra: Rings, Modules and Categories I" by Carl Faith

(5) Check out also the article entitled Noether Isomorphism theorem in the world heritage encyclopedia http://www.gutenberg.us/articles/noether_isomorphism_theorem Sbilley —Preceding undated comment added 23:04, 28 October 2016 (UTC)

Thank you for the response. I take the introduction to mean that you were not so much "tasked", as that you decided to take it upon yourself, inspired perhaps by the talk in question. As such, I would say that this needs to be discussed, probably at the Wikipedia talk:WikiProject Mathematics page, rather than simply done. If adopted, this would require a lot more changes than just in this page. I will also note that the talk said we should be using the name, not that we do; the page should reflect common usage. I'm at home, but I will look up in my bookshelf to provide specific instances. I think that the name "Noether's Isomorphism Theorems" is simply not widespread enough to warrant the kind of wholesale change you made. I support adding attribution, and perhaps even a section on the name, but this is a sort of advocacy-through-naming-in-Wikipedia that does not seem warranted. We can take it to the WikiProject, if you wish, given the low traffic here, before determining what the consensus is. Magidin (talk) 21:12, 29 October 2016 (UTC)

Here's from my bookshelf: Groups and Symmetry by M.A. Amstrong; "Isomorphism Theorems". Representation Theory of Finite Groups and Associative Algebras by Curtis and Reiner; "Fundamental Theorem on Homomorphisms". Abstract Algebra, by Dummit and Foote; "Isomorphism Theorems". Basic Algebra by Jacobson; "Fundamental Theorem of Homomorphisms of Monoids and Groups". Algebra by Thomas Hungerford; "Isomorphism Theorems". A course in Group Theory by John Humphreys; "Homomorphism Theorem". Universal Algebra by Grätzer; "Isomorphism Theorems". Algebra by Serge Lang; no name given. Groups and Geometry by Peter Neumann, Gabrielle Stoy, and Edward Thompson; "Isomorphism Theorems". An introduction to the theory of groups (4th Ed) by Joseph Rotman; "Isomorphism Theorems". A course in the theory of groups (2nd ed) by Derek Robinson; "Isomorphism Theorems". Advanced Modern Algebra (2nd ed) by Joseph Rotman; "Isomorphism Theorems". Elements of Algebra by John Stillwell; "Isomorphism theorem for groups". None of my books refer to them "Soether Isomorphism Theorems". Magidin (talk) 16:37, 3 November 2016 (UTC)

@Magidin: You're wrong about Robinson, look at the name of the section. Njsnyder (talk) 16:59, 31 October 2017 (UTC)

General case

In the second isomorphism theorem, it says that $[B]^{\Phi }$ is the collection of equivalence classes but what follows has no sense. It should probably be

$[B]^{\Phi }=\left\lbrace K\in A\mid \exists \,b\in B,\;\Phi \cap (K,b)\neq \emptyset \right\rbrace /\Phi$

Noix07 (talk) 16:22, 17 October 2018 (UTC)

Ah, now I get it, i previously thought the slash meant "so that". $K\in (A/\Phi )$ so that... Noix07 (talk) 16:29, 17 October 2018 (UTC)

File:Diagram_of_the_fundamental_theorem_on_homomorphisms

Hi D.Lazard, could you clarify what is not clear on the removed diagram: https://en.wiki.x.io/wiki/File:Diagram_of_the_fundamental_theorem_on_homomorphisms.svg at https://en.wiki.x.io/w/index.php?title=Isomorphism_theorems&oldid=976843241 ? An extremely verbose explanation can be seen at: https://math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426 I was no sure how much to write down on that caption. I believe this diagram can be much more useful to non-experts than existing diagrams on the page.Cirosantilli2 (talk) 11:47, 26 February 2021 (UTC)

If you use an arrow diagram, isomorphism must appear as arrows, and you must specify which are the exact sequences, which are the commutative triangles and squares. So you must have an exact sequence, say on a line,

$0\to \ker(f)\to G\to G/\ker(f)\to 0$

above the exact sequence

$H\leftarrow \operatorname {im} (f)\leftarrow 0$

connected by vertical sequences

$G\to H$

and

$0\to G/\ker(f)\to \operatorname {im} (f)\to 0.$

With such a diagram, the caption could be Expression of isomorphism theorem A in terms of an arrow diagram: the aligned sequences of arrows are exact sequences, and the square is a commutative diagram.

It is possible to orient the diagram differently, for having $G\to H$ horizontally. This has the disadventage to have the longest sequence vertically (however, because of the places of figures in WP, this may also be an advantage). The advantage is a better emphasize on the starting arrow. So, the orientation is a matter of taste.

Note that this diagram is valid only for additive (commutative) groups, vector spaces and modules. D.Lazard (talk) 17:41, 26 February 2021 (UTC)

By the way, the other isomorphisms theorems can also expressed and even proved in the same way. For example, for the theorem B, one can write the two exact sequences

$0\to S\to S\to 0,$

$0\to S\cap T\to S\oplus T\to S+T\to 0,$

the arrow $S\cap T\to S\oplus T$ beimg defined by $x\mapsto (x,-x).$ Then, the sequences are connected by natural vertical arrows, and the snake lemma gives the exact sequence (the part of the long sequence that is above the first sequence consists only of zero modules).

$0\to S\cap T\to T\to (S+T)/S\to 0.$

which is theorem B.

The other theorems can certainly be expressed similarly, but presently I do not remember which diagrams must be drawn.

This formulation of the isomorphism theorems deserve clearly to be explained in a separate section. D.Lazard (talk) 16:02, 27 February 2021 (UTC)

Clarification of universal algebra version of Thm B

Note that writing $C(A)$ for the congruences of $A$, the theorem says the following two ways of quotienting $B\hookrightarrow A$ by $\Phi \in C(A)$ give isomorphic results:

• Pulling back $\Phi$ to $C(B)$ by restriction, then quotienting in $B$ (this is $B/\Phi _{B}$)
• Pushing forward $B$ all the way to $A/\Phi$ and taking the algebra generated by the image (this is $[B]^{\Phi }$)

(not adding to page because it might violate WP:NOR, feel free to steal if I'm mistaken) 77.137.68.75 (talk) 19:36, 22 August 2023 (UTC)

Correction: the second point should read:

• Taking the quotient of any subalgebra of $A$ containing $B$ and contained in the closure of $B$ under $\Phi$ (this closure can be seen as the preimage of $B/\Phi$ under the quotient map)

77.137.68.75 (talk) 08:59, 27 August 2023 (UTC)