Talk:Wreath product

message: 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)
replies: text: 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)
thread_title:  Categorical definition 

message: Does someone know how to say "wreath product" in french ? 129.199.158.163 08:10, 7 November 2006 (UTC)
thread_title:  Wreath product in french 

message: 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)
replies: text: 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)
thread_title:  Style complaint 

message: 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)
replies: text: 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)
thread_title:  Minor Edits in Examples 

message: The article says :
"* The imprimitive wreath product action on Λ×Ω.
: If (aω,h)∈A WrΩ 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)
thread_title:  Imprimitive wreath product action on Λ×Ω. 

message: Just wondering. Article says that in general, wr product operation is not associative, which I believe, but I was reading here  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)
replies: text: 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)
thread_title: Associativity?

message: 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)
thread_title: Before Krasner & Kaluzhnin

message: The definition provided uses a group  acting on a set \Omega.  I'm not sure why it doesn't instead let  be a permutation group H\subseteq S_n. Then instead of having to go through defining A^\Omega we could simply use the group .  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 .  (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  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)
thread_title:  Overly complex definition? 

message: 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_1h_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)
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)
thread_title:  Definition of action of <math>H</math> on <math>\Omega</math> 

message: 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)
thread_title:  cardinality formula should hold for infinite sets too