Definition talk:Group Action

Hey Matt, I understand that $g*x~\mbox{or}~g\cdot x$ may be overused but the $ \wedge $ product is a pretty special symbol (as I'm sure you know). Perhaps a good idea to reserve it for logical "and" and the wedge product for Exterior Algebras. What do you think?--Grambottle 16:40, 17 December 2008 (UTC)

Oo-er, not too sure. I grew up with $\wedge$ and I've used it exclusively when developing the theory of group actions (as far as it goes, it's not far, but there's a fair few pages). I don't know - it would be some work to change it, and even then there may be those who say, "what happened to wedge?" We need a consistent approach but beyond someone just deciding to use a particular notation when introducing a subject, there's no hard answer.

What I tend to do is, when using an ambiguous symbol on any page nowadays, is define it on that page: "Let $g \wedge x$ be the group action of $g$ on $x$" or whatever, and everyone's happy. --prime mover (talk) 20:45, 17 December 2008 (UTC)

I go back and forth on this. At the moment, I think it probably makes the most sense to leave it as is but make sure that all pages that refer to group action have it referenced clearly to avoid confusion. I did a search for \wedge and got this, and these are the pages that link to the definition of group action. I'll try to remember to go through them this weekend, but I have an english paper I need to finish first. --Cynic-(talk) 23:06, 17 December 2008 (UTC)

I've gone through every occurrence of $\wedge$ and every time it is used as a group action, it has either been specified explicitly to be a group action, or has been provided with a direct link to the definition page.

Now for a few statistics from my "library":

a) Open University "Groups and Geometry" M336: uses $\wedge$. Date: 1994 (oh that's so last millennium ...)

b) Allan Clarke "Elements of Abstract Algebra" uses $*$. Date: 1970.

c) John Humphreys: "A Course in Group Theory" uses $\cdot$, doesn't call it a "group action" but instead refers to a $G$-set. Date: 1996 (okay I concede, it's later than the one from 1994 above).

I was sure I had more books which discuss group actions. Seth Warner's "Modern Algebra" (1964) doesn't seem to mention the concept (and relegates all the important results about groups, including the Sylow Theorems to exercises at the end of the chapter).

Ian Stewart's "Galois Theory" (3rd Edition, 2004) surprisingly doesn't seem to introduce the concept (but then I haven't studied it thoroughly) and introduces the Sylow theorems in an appendix.

I was sure I had more books on group theory than that. Whatever.

IMO it's better to have a more "special" symbol for a group action than the amorphous ubiquitous $*$ or $\cdot$ or juxtaposition, because then there's far less chance of confusing the "group action" with the "group product". Enough groups are introduced with the operation $*$, and $\cdot$ is conventional for the product of numbers, so introducing group actions on such groups leaves you open to facing serious ambiguity charges, which is worse than public intoxication.

But what do I care, it's not my website. Do what you want to do, as long as you don't tell me to go through and change everything I've done, I'd rather drink sick. --prime mover (talk) 08:15, 20 December 2008 (UTC)

I have slightly rewritten the first defining line so as to avoid the ugly $X = g * x \in X$ part. If appreciated I will adapt it on all of PW. --Lord_Farin 03:27, 6 June 2012 (EDT)


 * (the discussion that was here has moved to Definition talk:Boolean Algebra. --Lord_Farin (talk) 09:05, 23 August 2012 (UTC))~

Permutation representation
The second definition is that of a permutation representation. I'll try to find a reference for that term. A group action determines a permutation representation and vice versa. --barto (talk) 08:18, 15 July 2017 (EDT)


 * Done. See Definition:Permutation Representation. We can add a remark that a group action is sometimes defined as the permutation representation itself, but that we make a clear distinction. --barto (talk) 08:37, 15 July 2017 (EDT)