Talk:Isomorphism theorem

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Field: Algebra

A discussion

I think we're seeing a bit of an over-reaction to the idiotic practice of using "cats" rather than "cat" as the title of an article about cats. This is not an article about a general concept of an isomorphism theorem, of which the reader may be expected to encounter thousands of concrete cases, and another reader may encounter thousands of others that the first reader doesn't run in to. Rather, it is an article about three particular theorems that often are mentioned in the same breath. So "isomorphism theorems" would seem appropriate as the title. Comments? Michael Hardy 01:12 Mar 21, 2003 (UTC)

It would make sense to use Isomorphism theorems as the title -- and it would make sense to use Cats as a title instead of Cat. The singular convention is to make linking easier within text, and that applies in this title as well. Hypothetically:

This result is a consequence of the first isomorphism theorem for groups.

Still, it's not a big deal to violate this for a good reason, if there's a redirect.

But as for the reason, there are indeed thousands of different concrete cases for the general concept of isomorphism theorem: three for groups, three for rings, three each for Rmodules for each ring R, etc. There are not just three isomorphism theorems. True, they can all be summarised into just three theorems that begin "Let C be a category such that ...", but of course that's not how we want to state them in this article -- at least not at first.

Thus I would agree with you in some cases, as long as there was a redirect from the singular, but not in this case.

-- Toby 06:27 Apr 1, 2003 (UTC)

Oops .... I had that backwards. I meant: the idiotic practice of using "cats" as a title rather than "cat". I think the singular convention usually is the right thing. But in some cases, e.g., Joint Chiefs of Staff, The Beatles, and Legendre polynomials, the plural is the right way to do it. Michael Hardy 18:34 Apr 1, 2003 (UTC)

I agree with you for the first two examples -- that's a good reason. As for the last example, however -- there are thousands of Legendre polynomials, infinitely many in fact ^_^. -- Toby 05:40 Apr 13, 2003 (UTC)

How many there are, as long as it's more than one, is not the essential point; the essential point is that no one seems interested in any particular Legendre polynomial except because it is one member of that sequence. Michael Hardy 21:55 Apr 13, 2003 (UTC)

Yes, hence my smile. You had implied above that the number was relevant with your talk of "thousands" -- but of course what you have just now is the truly relevant aspect. -- Toby 12:02 May 14, 2003 (UTC)

Are we missing an isomorphism theorem here? My Dummit and Foote Abstract Algebra lists a fourth "Lattice Isomorphism Theorem" which states that the lattice of a quotient group G/H corresponds directly to the lattice of subgroups of G containing H. Also D&F calls the second isomorphism theorem the "Diamond Isomorphism Theorem" so if people know of other sources that use this name maybe we should include it.

I've seen it published both ways, but isn't it far more conventional to list the second and third in the opposite order of this article? Nnn9245 21:45, 31 May 2006 (UTC)

It seems like including the commutative diagrammes that go with each theorem would be very useful in visualising what they say. If all I had were these descriptions, and I weren't otherwise familiar with the isomorphism theorems, they would look very difficult. But with the commutative diagrammes, they become almost obvious. Mraj 15:33, 11 October 2006 (UTC)

Pictures!

This article is in desperate need of some more pictures. Each theorem should have a nice diagram to go with it. asmeurer (talk | contribs) 00:12, 28 October 2012 (UTC)

History

The section "History" seems vague to me. Precise dates should be given. For example W. Burnside, Theory of Groups of Finite Order, 2 éd., 1911, repr. Dover, 2004, p. 41, gives as an exercise : « If H, h are self-conjugate [i. e. normal] sub-groups of G, and if h is contained in H, so that H/h is a self-conjugate sub-group of G/h, shew that the quotient of G/h by H/h is simply isomorphic [i. e. is isomorphic] with G/H. » This is he third isomorphism theorem. When was it stated for the first time ? Marvoir (talk) 17:35, 1 November 2012 (UTC)

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)