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Vipul 14:23, 30 May 2007 (IDT)
Group action property space
Here's what I'm unsure about in group action articles with respect to property theory: I'm not sure what the property space is exactly. There are two possible notions of group actions being similar: the one that I mentioned in an article, and a related one where the two groups are the same and the automorphism within the group is trivial (ie, on is similar to on if there is a bijection such that ). The second one is tighter than the first, and it implies a different property space: things can be organized by classes of 'strong' or 'weak' similarity. Both are useful, actually. For example, every non-primitive transitive group action is similar to a subgroup of a particular wreath product of parts of the group, acting on a particular coordinatization of the set acted on corresponding to block structure. This relies on the weak similarity notion. On the other hand, within the stronger similarity, it's true that every regular action is similar to the right regular action (these are definitions that I need to write up). So, I'm not sure how the property space should be described. Also, I'm not sure how to phrase statements like this in property theory: "something with this property is isomorphic to something else that we can characterize in a particular way." I'd appreciate advice with these things. I think this is a really interesting project.
LittleDan 01:27, 9 November 2008 (UTC)
Well, in the general case, I guess I'd lean towards a stronger sense of equivalence, but I don't really understand how that would work as something to base a property space around, when I think about it. It seems like this could only work something like Subgroup#Equivalence_of_subgroups. Here, you decided that two subgroups of the same group are equivalent if there's any automorphism of G taking H_1 to H_2. To me, this seems like the only consistent way to phrase it. The stronger notion of group action equivalence makes sense if we're considering multiple actions on one group, but I don't think it's well-formed as a property space in general, since group/action pairs up to some kind of equivalence are what's needed. So, would it then be reasonable to say, in the group action article, that there are two notions, equivalence and similarity, and the group action property space is built around similarity but when studying a particular group, (strong) equivalence is what should be considered? LittleDan 23:54, 9 November 2008 (UTC)
OK, thanks! I'll be sure to make this distinction clear in what I write. LittleDan 08:03, 10 November 2008 (UTC)