Talk:Wreath product

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Contents 1 Transitivity required? Direct Product vs Sum 2 Minor correction: specify permutation action 3 Categorical definition 4 Wreath product in french 5 Style complaint

[edit] 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.

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)

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)

[edit] 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,

C_{2 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.

[edit] 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)

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)

[edit] Wreath product in french

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

[edit] 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)