Talk:Wreath product

Transitivity required? Direct Product vs Sum

It seems like all we need is that H acts on the set U. Is it necessary that H act transitively, or that it be a permutation group?

Also, if U is infinite, I have an inkling that we should use the direct product rather than the direct sum of copies of G. Although I don't know why. 199.17.238.92 22:42 Nov 13, 2002 (UTC)

All my references seem to require H to be a transitive permutation group for the unrestricted product; but it appears that the generalized wreath product does not require transitivity; I don't have a written reference, but you can try [1] for example (it seems pretty dense going).

If H is not a permutation group, but only a transformation group, then I think we just find that if we define N = {h in H : h.u = u for all u in U}, then we essentially get G wr (H,U) = (G wr (H/N, U)) × N, with (H/N, U) a permutation group; so it's typically not very interesting.

"Permutation group" and "transformation group" mean one and the same thing: a group which acts on a set! Mfluch (talk) 21:25, 26 August 2010 (UTC)[reply]

I think the reason for the restriction to transitive H is one of applications. We certainly never need to worry about transitivity when talking about G wr K for ordinary groups (i.e., taking K as the permutation group (K, K)); and one writer [2] seemed to imply that the wreath product was originally developed as a tool to describe the structure of permutation groups, where intransitive groups are just direct products or semidirect products of transitive subgroups. But the real answer is - I dunno!

As regards direct sum, we actually want the external direct sum; which is a subset of the direct product over U, including only those elements whose components are e_G except for a finite number of components. Particularly for uncountable U, this allows us to ensure that limits of a product exist, etc. The construction is not unlike the Tychonoff product of topologies. Chas zzz brown 00:48 Nov 14, 2002 (UTC)

There are two wreath products, the unrestricted (also called standard or regular) wreath product A Wr (H,U) and the restricted wreath product A wr (H,U). In the construction of the unrestricted wreath product one uses the direct product, in the construction of the restricted wreath product one uses direct sum. If U is a finite H-set, then the unrestricted and restricted wreath product agree.

In this article both, the restricted and unrestricted wreath product are used and mixed up. For example, the lamplighter group is an restricted wreath product whereas the Universal Embedding Theorem (every extension of A by H is a subgroup of A Wr H) requires the unrestricted wreath product (it is false otherwise).

This article definitvely needs to be rewritten (more or less from the scratch) due to all this confusion and if I find some time I will do this soon. Mfluch (talk) 21:25, 26 August 2010 (UTC)[reply]

For transformation groups (=permutation representations=group actions), the wreath product does not decompose as you described (five years ago). For instance if G=1, H=C4, N=C2, then G wr (H,H/N) = (G x G) x| H = H has an element of order 4, but (G wr (H/N,H/N)) x N = ((G x G) x| H/N) x N = C2 x C2 has no element of order 4.

The restriction to transitive groups is a bit nonstandard, and I'll probably take care of that in the answer to the answer to "Style complaint" below. JackSchmidt 20:04, 30 November 2007 (UTC)[reply]

Minor correction: specify permutation action

Minor correction to the article proposed: I think the example should be C₂ wr (S_n, n) instead of C₂ wr S_n, if we want to keep the notation consistent. Could anyone of the experts in the subject change it or explain why not? Regards, Alex.

Alex - I don't think there is any ambiguity in this case since n is given to specify that H is a subgroup of

S_{n}

, which in the case

H=S_{n}

is obvious. - Gauge 22:00, 23 Feb 2005 (UTC)

I agree with Alex, the page says "Finally, since every group acts on itself transitively, we can take U = H" so it clearly imply that the default is to take |H| copies of G. According to the current page,

C₂ _wr S_n stands for C₂ wr (S_n, S_n) and not for C₂ wr (S_n, n) or more précisely C₂ wr (S_n, [1..n]). I made the suggested change.

Categorical definition

I've come across the wreath product for categories, and the definition seems to be a more general case. I'm wondering if it's worth expanding on here?

Also, the definition is not the easiest to read. How about using polynomials as an analogy (I know it's not perfect, but it does give one a rough idea of what it is) -3mta3 13:30, 7 August 2005 (UTC)[reply]

Please write the categorical version up. I haven't seen it before and it would be worth putting here. - Gauge 00:09, 11 August 2005 (UTC)[reply]

Wreath product in french

Does someone know how to say "wreath product" in french ? 129.199.158.163 08:10, 7 November 2006 (UTC)[reply]
The french translation is "produit en couronne".

Style complaint

The definition should not include an example. Really this page should be rewritten somewhat. —Preceding unsigned comment added by 130.195.86.40 (talk)

It definitely needs to be rewritten as the page completely mixes and confuses unrestricted and restricted wreath products. Factually in the current form the statements of this page are inconsistend and false. Mfluch (talk) 19:43, 25 August 2010 (UTC)[reply]

The distinction between restricted and unrestricted should only be made after a definition is given for actions on finite sets — Preceding unsigned comment added by 137.54.149.138 (talk) 21:22, 7 February 2012 (UTC)[reply]

Minor Edits in Examples

I replaced the slightly unusual notation Cp by the more usual Zp and gave the reference to the Wikipedia page on cyclic groups.

I can not figure out why the Zp in the reference "where Zp is the cyclic group of order three" came out in smaller font than the other Zp's. It was not intentional. Oh well, better to have the reference in the wrong font than not to have the reference at all.

It took me the longest time to figure out what "Cp" is, despite the fact that I have 50 years experience working in mathematics. I thought I'd save other the readers the same trouble.

I realize that "Cp" is the more logical notation, as is hinted at in the Wikipedia page on cyclic groups. It seems to me that it is foolish to try to change what is (thanks to Bourbaki and "Zahlen" for Z) at last a standard notation for Z/pZ. [Come to think of it, Z/pZ might be even better.]

Instead of Z for the integers (and Zp for the cyclic groups of order p), the older texts use a wide variety of notation. Let's be thankful for the progress that's been made and not try to turn it over at this time.

If you MUST use the Cp notation, please explain why you are using it (e.g., "because the Cp is the usual notation for Zp when symmetry groups of polyhedra are considered," or "the use of Cp instead of Zp is intended to emphasize that the cyclic groups of order p, though abelian, are written multiplicatively instead of additively in the context of wreath products," or whatever your reason is). Surely, you don't plan to overturn the vast majority of usage in the current mathematical literature with one sparse section of one Wikipedia page.

And, if you use the "undo" button, please at least add something like "where Cp is the cyclic group of order three," so that the average reader will know what you are talking about.

DeaconJohnFairfax (talk) 00:56, 1 July 2008 (UTC)[reply]

In mathematics Zp is used when the group of p elements is written additively and Cp is the common notation of the group with p element when written multiplicatively. Both groups are identicaly (that is, isomorphic), only the law of composition is uses a different symbol. Mfluch (talk) 02:25, 28 August 2010 (UTC)[reply]

Page rewritten

I've just rewritten the page more or less from scratch. The rewrite process is not yet complete and there might be still some minor mistakes. However, I hope that the current version is more consistent, especially when it comes to the distinction of restricted and unrestricted wreath product, which is essential when the H-set $\Omega$ is not finite. --Mfluch (talk) 12:19, 29 August 2010 (UTC)[reply]

Thanks. The permutation actions have been more or less deleted, and so should be restored. The choice of references is less than ideal. Standard material should be sourced to textbooks, and some effort needs to be made to only source true statements. Several examples have been deleted, destroying some reasonable two-way linking that previously existed. The inline tex violates the manual of style, and will probably look better just using wiki markup, A Wr_Ω H, etc. JackSchmidt (talk) 20:16, 29 August 2010 (UTC)[reply]

I know that it is at the moment still sub-obtimal. But there was so much confusion in the previous version that I didn't see how to resolve it. In the previous version there was no references at all (except one)...Mfluch (talk) 09:47, 30 August 2010 (UTC)[reply]

Inline math style changed as suggested. The permutation actions will definitely be restored. I just didn't have time yet to write the details out. I will work on it.Mfluch (talk) 10:37, 30 August 2010 (UTC)[reply]

Yup, you did good. I obviously didn't have time to restore them either (or any of the other fixes). Just making a todo list and letting you know your edit had been reviewed and de-facto accepted by at least one other editor. It's over-all an improvement. – Also all your new edits since my first comment are very good. The todo list is more or less done. I'll "fix" references if I get time: there is a nice description of conjugacy classes and irreducible representations of wreath products of finite groups that is notable and easily sourced, and if I do that I'll probably stumble upon my bibliography for wreath products (mostly of finite groups). I think Robinson's textbooks might have some very good examples of wreath products, like Z wr Z is finitely generated soluble, but not noetherian or polycyclic. JackSchmidt (talk) 21:26, 30 August 2010 (UTC)[reply]

Yes, it would be nice to have some more references. I think some of the references I have places are still suboptimal. Well, I think there are still many things one can improve, but I hope the rewriting I have done has been useful...

I hope that the newly created section on Notation and Conventions describes the issue correctly (but please correct me if I am wrong). Mfluch (talk) 16:21, 31 August 2010 (UTC)[reply]

Imprimitive wreath product action on Λ×Ω.

The article says :

"* The imprimitive wreath product action on Λ×Ω.

If (a_ω,h)∈A Wr_Ω H and (λ,ω')∈Λ×Ω, then

(a_{\omega },h)\cdot (\lambda ,\omega '):=(a_{\omega '}\lambda ,h\omega ')

."

If my calculations are right, this is not an action of the wreath product on Λ×Ω.

I think that the right definition is

(a_{\omega },h)\cdot (\lambda ,\omega '):=(a_{h\omega '}\lambda ,h\omega ')

.

I derived it from Rotman's "permutation version" of the wreath product.

By the way, wouldn't it be more accurate to write

(\ (a_{\omega })_{\omega },h)\cdot (\lambda ,\omega '):=(a_{h\omega '}\lambda ,h\omega ')

(with $\ (a_{\omega })_{\omega }$ instead of $\ a_{\omega }$ in the left member ? Marvoir (talk) 12:56, 8 February 2011 (UTC)[reply]

Associativity?

Just wondering. Article says that in general, wr product operation is not associative, which I believe, but I was reading here [3] that wr products are associative up to some isomorphism and even fully associative if the cartesian product is treated as associative. Is this worth mentioning? 59.101.46.39 (talk) 20:00, 25 October 2011 (UTC)[reply]

Yes, wreath products are associative. The rewrite deleted this sourced statement and replaced it with its unsourced negation. Of course one has to be careful with which action one uses, but the imprimitive action works quite well. I don't like the notation being used (and someone above thinks it might be wrong), so I won't fix it today. JackSchmidt (talk) 20:22, 1 November 2011 (UTC)[reply]

Before Krasner & Kaluzhnin

The following article provides some pre-history of the wreath product:

Mario Petrich (1970) "Biographical Note", Semigroup Forum 1(1): 184

Petrich cites Frobenius & Shur from 1906 and Specht from 1933, before a Comptes rendues (fr) source for Krasner & Kaluzhnin from 1948. — Rgdboer (talk) 21:12, 18 August 2019 (UTC)[reply]

Overly complex definition?

The definition provided uses a group $H$ acting on a set $\Omega$ . I'm not sure why it doesn't instead let $H$ be a permutation group $H\subseteq S_{n}$ . Then instead of having to go through defining $A^{\Omega }$ we could simply use the group $A n$ . The group action can then be defined as $A^{n}\rtimes _{\varphi }H$ where $\varphi (h)=h^{-1}$ or to put it more concretely:

$(g_{0},h_{0})\times _{A\,\wr _{n}H}(g_{1},h_{1})=(g_{0}\times _{A^{n}}h_{0}^{-1}(g_{1}),h_{0}\times _{H}h_{1})$

This is the approach taken in McMahan (2003)^[1]. (Although they use a different formulation since they don't incorporate the semidirect product.) I think that this approach is obviously clearer, however it does on face value seem to be less general, since in the existing definition $H$ is any symmetry group, whereas in the proposed definition it's specifically a permutation group. However by Cayley's theorem every group is isomorphic to a permutation group, so it is trivial to substitute any symmetry group for a permutation group, and there are already groups that aren't defined as symmetry groups, so it's not like substitutions didn't have to be made already.

I really think we ought to use such a definition, but it's possible I'm overlooking some value in the existing definition. AquitaneHungerForce (talk) 10:13, 14 July 2022 (UTC)[reply]

References

^ McMahan, Peter (2003). Wreath-Product Polytopes (PDF) (Thesis). Reed College.

Definition of action of $H$ on $\Omega$

It is stated that

h\cdot (a_{\omega })_{\omega \in \Omega }:=(a_{h^{-1}\cdot \omega })_{\omega \in \Omega }

which leaves me with the following problem

(h_{1}h_{2})\cdot (a_{\omega })_{\omega \in \Omega }=(a_{(h_{1}h_{2})^{-1}\cdot \omega })_{\omega \in \Omega }=(a_{{h_{2}}^{-1}{h_{1}}^{-1}\cdot \omega })_{\omega \in \Omega }=h_{2}\cdot (a_{{h_{1}}^{-1}\cdot \omega })_{\omega \in \Omega }=h_{2}\cdot h_{1}\cdot (a_{\omega })_{\omega \in \Omega }

Where does the used notation come from? In Bhattacharjee (1998)^[1] the action is from the right, hence inversion is actually needed. I don't see that it is needed here. Hvtka (talk) 11:44, 15 November 2023 (UTC)[reply]

cardinality formula should hold for infinite sets too

The article states that the formula |A≀_ΩH| = |A|^|Ω||H| holds if all three sets are finite. But it seems to me that this formula will hold for all sets, as long as it's clear that the unrestricted product is meant. It is, after all, as a set, just a product of a power of sets. I think an analogous formula for restricted wreath product is also obvious. The upshot is, there seems to be no need to assume finiteness. -lethe ^{talk + contribs} 01:48, 25 November 2023 (UTC)[reply]

^ Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998), "Wreath products", Notes on Infinite Permutation Groups, Lecture Notes in Mathematics, Berlin, Heidelberg: Springer, pp. 67–76, doi:10.1007/bfb0092558, ISBN 978-3-540-49813-1, retrieved 2021-05-12

[1] McMahan, Peter (2003). Wreath-Product Polytopes (PDF) (Thesis). Reed College.

[2] Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998), "Wreath products", Notes on Infinite Permutation Groups, Lecture Notes in Mathematics, Berlin, Heidelberg: Springer, pp. 67–76, doi:10.1007/bfb0092558, ISBN 978-3-540-49813-1, retrieved 2021-05-12

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