Talk:Group with operators

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 Field: Algebra

I don't understand the significance of the remark about a group with operators being a mapping from Omega to the set of group endomorphisms. Isn't Omega just a distinguished subset of endomorphisms in the first place so that the mapping is inclusion? Or is there a more general notion of group with operators which is being alluded to? --Michael Kinyon 00:38, 31 July 2006 (UTC)

Agreed. To say you have an object in this functor category is to pick-out a group and to map functorially monoid elements (= hom-set of category M) to elements of hom-set Hom_Grp(G,G)=End_Grp(G). So it takes e in M to id_G, and products to compositions. OT1H, in my 1974 ed, Bourbaki (I, Sect.4.2) puts no constraint on whatsoever (it's just a set). OTOH, a monoid must contain the identity and be closed under composition. So the Bourbaki Omega cannot precisely be identified with the functor category's M. One must expand Omega to the smallest monoid containing it, or view the mapping as an inclusion. I've added some material to try to clarify, but this needs some work (and a citation/ref). Purfideas (talk) 01:23, 11 May 2012 (UTC)
Found explicit use of the functor category definition in Mac Lane, so added the ref. Purfideas (talk) 02:36, 11 May 2012 (UTC)

stabilizer-like structure[edit]

Does somebody know something related to the following structure:

Let E be an R-module, A \subset 2^E, B\subset 2^R, and
E(A,B) = { x\in E | \forall a\in A, \exists b\in B: b x \subset a }

(Note, x is an element, but a,b are subsets of E - sorry for the ascii-limited notation; I wrote 2^E for the power set although I hate this notation.)

It seems like something in-between a stabilizer, radical of a ring, and the von Neumann/Kolmogorov definition of bounded subsets in TVS / modules. I think this should be considered somewhere, but could not find it anywhere. — MFH:Talk 22:03, 12 October 2006 (UTC)

Typo in the ring example?[edit]

I think it must be a typo to say "operates on the operator domain". I cannot find this terminology in Bouraki (the one reference at the time of this writing). Clearly scalar multipliation is an endomorphism of vector spaces (it's a diagonal matrix after all), and similarly for R-modules. So this must mean R is operator domain for underlying Abelian group of M. I'm going to clarify accordingly. Please roll it back with a reference to this terminology if I'm wrong. Purfideas (talk) 01:36, 11 May 2012 (UTC)

It looks like it used to say the group operated on the ring? What a weird typo. I think the steps you have taken look good. Rschwieb (talk) 17:24, 11 May 2012 (UTC)

Set or family[edit]

The article says that "the algebraic structure group with operators or Ω-group is a group with a set of group endomorphisms".

I replaced "set" with "family" and gave this reason : " "Family", not "set". With a set, it is impossible to define a homomorphism between groups with operators."

Rschwieb reverted, with this comment : "previous version was simpler and correct, and reason to change does not appear to be valid. Pls discuss on talk, if necessary."

Well, the reason I gave is not the good one. The true reason is explained by J.S. Rose : "Clearly, any set of endomorphisms of G is a suitable operator domain for G. However, in 7.27 we do not restrict Ω to be a set of endomorphisms of G since we wish to allow the possibility that there are distinct elements of Ω which operate on the same way on G." (A course on group theory, 1978, repr. Dover, 1994, p. 138. I hope that it is clear enough. Marvoir (talk) 18:16, 1 November 2012 (UTC) (Message edited.) Marvoir (talk) 18:52, 1 November 2012 (UTC)

The main thing seems to be that most sources use "set," and "family" is not nearly as formally defined as "set" in most contexts. I cannot see any reason at all to use "family." Moreover, the link family (mathematics) is useless compared to set (mathematics). Rschwieb (talk) 20:30, 1 November 2012 (UTC)
I'm French speaking, and in French we say "une famille" where English speaking people say "Indexed family". Okay, thus, I had better said "an indexed family". I see that in the body of the article, you replaced my "family" with "indexed set". For "indexed set", there is a redirection toward "Indexed family"...
Now, if really Ω is always a set of endomorphisms of G (as it was stated explicitly in the former version, but, I don't know why, you didn't remove my correction of this passage), then G is determined by Ω (if Ω is not empty). How then can two different groups have the same operator domain, as assumed in the definition of an endomorphism between two groups with operators : "Given two groups G, H with same operator domain ..." ?
And how do you explain the warning of Rose ? W.R. Scott says the same thing : "However, S cannot be considered as a set of endomorphisms of G, since distinct elements s and s' of S may induce the same endomorphism." (Group Theory, Dover reprint, p. 40.) Marvoir (talk) 07:54, 2 November 2012 (UTC)
P.S. In the comment about your last edit, you wrote : " I am not even sure if anything but sets are used for this theory, so it seems safest to say "set." "
I have quoted two authors who insist that the operator domain is not necessarily a set of endomorphisms and can not necessarily be viewed as such. You don't quote any author who defines a group with operators as a group with a set of endomorphisms. Thus, your definition in the beginning of the article is cetainly wrong. It becomes accurate if one replaces the word "set" with "indexed family". It is perhaps true that "indexed family" is not used by the authors in the definition of a group with operators, but it is the one way to make your (initial) definition accurate without changing it entirely : the definition of a group with operators as a group with a "set" of endomorphisms is false and explicitly prohibited by Rose and by Scott. In fact, most, and perhaps all, authors define a group with operators as a group with a distributive action of a set on the group. Classically, an action of a set A on a set B can be viewed as an application of A into the set of the applications of B into itself, and, in turn, an application of A into the set of the applications of B into itself can be viewed as an indexed family (with index A) of applications of B into itself.
If you don't like to speak about an indexed family in the initial definition, you mus replace this definition by a correct one, and then I see only the definition as a group wih a distributive action of a set on this group. (Edited) Marvoir (talk) 08:31, 2 November 2012 (UTC) (Edited.) Marvoir (talk) 09:59, 2 November 2012 (UTC)
  • You are obviously interpreting your sources incorrectly. The way the operators operate on G has nothing to do with the set nature of Omega.
  • I have no idea why you are using Rose as a source. The googlebooks preview clearly shows Rose using "set".
  • If you googlebooks for "group with operators," your first hit will be Bourbaki, which uses "set."
  • It is patently absurd that it would not be a set. The endomorphisms are a subset of the set End(G).
  • Is there still disagreement?

Rschwieb (talk) 13:49, 2 November 2012 (UTC) Addendum: I see your point about "indexed set" being a redirect here. In that case, I think you're right we should use "indexed family" for that link. Rschwieb (talk) 13:58, 2 November 2012 (UTC)

Of course, these authors speak of Ω as of a set. Where did I say that it is not a set ? But they don't speak of Ω as of a set of endomorphisms. You, however, in the initial definition, you say that a "group with operators or Ω-group can be viewed as a group with a set Ω of group endomorphisms". That is false. Rose says explicitly : "we do not restrict Ω to be a set of endomorphisms of G". And Scott says the same thing. Yes, to each element of Ω, there corresponds an endomorphism of G, but that doesn't implie that Ω is a set of endomorphisms. In fact, the article cites the exemple of a R-module, where the operator domain is the ring. An element of the ring is usually not a goup endomorphism of the addititive group of the module. Is it clear ? Marvoir (talk) 15:30, 2 November 2012 (UTC)
OK, now we finally arrived at where we are not understanding each other. I was convinced you were just replacing "set" with "family." Please understand that this is a sort of nuisance we editors come up against frequently. Sorry to have missed your point.
I didn't believe anyone would bother distinguishing between operators that effect the same transformation, but in retrospect that seems completely natural, because, for example, that is exactly what happens when you use conjugation action by central elements of G on G.
To be honest the length of your previous response discouraged me from taking it seriously. Usually reasonable changes have succinct explanations. Next time I go off the rails (or if you have a similar exchange with someone else) be sure to read what they write and try to figure out if they're talking about the same thing as you.
Is the current edition of the article satisfactory to you? Rschwieb (talk) 18:03, 2 November 2012 (UTC)
Another reason I didn't take you seriously was that the edit summary did not describe the mistake: With a set, it is impossible to define a homomorphism between groups with operators. All along I have been trying to convince you that the operators are a "set", but your point was that they are not a "subset of endomorphisms". Rschwieb (talk) 18:09, 2 November 2012 (UTC)
OK, we understand eachother, now. Perhaps I was not clear enough. The last form of the article seems satisfactory to me. As you pointed out, the expression "indexed family" is not used in a definition of a group with operators, thus it is better to avoid it in the beginning of the article. Marvoir (talk) 18:22, 2 November 2012 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── I like the current edition also. I'm glad we arrived here with only a few sparks flying around! :) Rschwieb (talk) 19:26, 2 November 2012 (UTC)