Talk:Group action

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I have added a reference to results of Higgins and me on the fundamental groupoid of an orbit space since this is a powerful result, and I hope people will find this useful. Ronnie Brown Sept 29, 2006

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I'm not sure about the standard terminology here:

• Is it called "faithful" or "free"?
• Is "invariant" and "stable" really the same?
• Are the sets G.x really called "traces"? --AxelBoldt

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• It's called "faithful", or "effective". ("Free" means that only the identity element has a fixed point.)
• I'm not sure about this one. I would use "invariant" for the sense you were talking about.
• The sets G.x are usually called "orbits" (as in the Orbit-Stabilizer Theorem).

--Zundark, 2001 Oct 28

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I removed the reference to permutation groups in the first paragraph since a permutation group on M is a subgroups of Sym(M), while a transformation group G on M is given by a (not necessarily injective) homomorphism G → Sym(M). So they are not the same. AxelBoldt 17:45 Oct 31, 2002 (UTC)

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If G acts on A, I think that "invariant" means G.A=A while "stable" means that G.A is a subset (possibly proper) of A.

If so, you can't actually have G.A ≠ A, with G a group. Charles Matthews 16:21, 15 November 2005 (UTC)

due to the neutral ? You're true. It you think to the whole matricial group (M) acting on othogonal matrices (O), the group O is not stable because MO is not an orthogonal matrice in general. Then my initial bracket "possibly proper" is actually not necessary.

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There's now an overlap with orbit (mathematics).

Charles Matthews 18:52 24 Jun 2003 (UTC)

I've now merged in that stuff. Also orbit (group theory) now redirects here. -- Fropuff 17:51, 2004 Aug 23 (UTC)

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In the first sentence under definition, g is used both as the group action, and as an element of the group G. Suggest use (say) α for the group action instead. Tveldhui 01:38, 17 January 2006 (UTC)tveldhui

the dot denotes the action, g is always a group element. -MarSch 16:25, 1 February 2006 (UTC)

Contents 1 Reference 2 n-transitivity 3 similar structure 4 sharply transitive = regular (= simply transitive) ?

[edit] Reference

I've added a reference from the group article which I believe is sufficiently familiar and encylopedic to satisfy many people. I propose we use this reference to obtain a uniform set of notations and terms, perhaps even in multiple articles. Perhaps I'll get around to it in a week or so, with consensus, if no one else does. Orthografer 18:53, 29 June 2006 (UTC)

[edit] n-transitivity

Hello,

you define n-transitivity to be "transitivity on Xⁿ", by which I suppose you mean the action on Xⁿ with

But then n-transitivity can not be defined as transitivity on Xⁿ, because you can never find a g satisfying

g(a,a) = (b,c)

for from X.

A correct definition would read: For pairwise distinct and there is a g such that

gx_k = y_k

holds for .

OK, seeing that both the x_k and the y_k have to be pairwise distinct (otherwise g would not be a bijection) we might restrict the action of Xⁿ to the subset of Xⁿ containing no multiple entries. If that is what you mean by "transitivity on Xⁿ" I guess it deserves being pointed out explicitly.

—The preceding unsigned comment was added by FarSide (talk • contribs) 08:23, 7 July 2006 (UTC2)

[edit] similar structure

Does anybody know the following structure, similar to the stabilizer, defined, for A,B\subset 2^G (the power set), by

G(A,B) = { x\in G | \forall a\in A \exists b\in B : b x \subset a }

(Note: a,b are subsets of G !)

Might this exist in the context of topological groups (where A,B would be neighborhoods of the identity element) ? It ressembles to the Kolmogorov definition of bounded sets

I ran across this in the context of modules over a ring (G would be a module over R, and B\subset 2^R). This seems so basic (and has nice properties) that I'm sure it already exists in literature, but I could not yet find it. Thanks in advance ! — MFH:Talk 21:50, 12 October 2006 (UTC)

[edit] sharply transitive = regular (= simply transitive) ?

As far as I understand the term sharply transitive is the same as regular or simply transitive, isn't it? If so, it would be good to mention this in the article. Florianhe 21:19, 11 January 2007 (UTC)