Talk:Group action

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The contents of the Properly discontinuous action page were merged into Group action on 8 February 2016 (UTC). For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page.

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"In mathematics, an action of a group is a way of interpreting the elements of the group as "acting" on some space in a way that preserves the structure of that space."

Unfortunately, introducing a concept (group action) by using one of the words in the concept (action) — in quotes, yet ("acting") — is worse than unhelpful. It is totally exasperating to anyone trying to understand the concept. It is the last thing we need in Wikipedia.2602:306:CF5D:1270:4023:7D48:4830:C217 (talk) 20:11, 27 August 2017 (UTC)[reply]

I reworded the first sentence in the lead. I think this is more clear. In a mathematics article, describing a concept in simply understood terms, before the technical definition is made, is often difficult.—Anita5192 (talk) 21:54, 27 August 2017 (UTC)[reply]

An editor recently changed all the center dot notations to baseline dots for the sake of uniformity in the article. I reverted, claiming that this was the wrong choice, with the intention of switching all the baseline dots to centered dots. Looking at the page I realized that this would be a mammoth job and the thought occurred to me that this might just be an English variant issue, and I should revert myself. All the references that I have checked either use a center dot or simple juxtaposition for the group action. I am not claiming any type of expertise here and am seriously asking where the baseline dot notation may be common? --Bill Cherowitzo (talk) 16:39, 18 September 2017 (UTC)[reply]

Apparently the baseline dot notation was introduced in 2012, see diff. I have never seen this notation outside of this article, but that doesn't mean much. Tea2min (talk) 09:36, 19 September 2017 (UTC)[reply]

I've seen it used (and used it myself a long time ago) but I've always assumed that it was only out of laziness or not knowing the use of \cdot in latex. So it would not be out of place for this article to do so :) jraimbau (talk) 14:50, 19 September 2017 (UTC)[reply]

I am thinking that I have seen this notation before and the mental image I have is of some older typewritten monographs (pre-TeX, when you used to have to hand off your manuscript to a secretary to type). My feeling is that if there is some segment of the population that regularly uses this notation, we should keep it (and the 2012 introduction into this article makes me think that this might be the case), but if it is only of nostalgic value then by the "principle of common usage" (don't go looking for a policy by that name, its my own terminology) we should change this to centered dot notation.--Bill Cherowitzo (talk) 17:15, 19 September 2017 (UTC)[reply]

I think the period is only used in older texts, so we should probably go with centered dots or juxtaposition. I lean towards the former, because using juxtaposition for both the group operation and the action might confuse newcomers. I have no strong opinions on the matter though; I would only argue that we should be consistent throughout the article. AxelBoldt (talk) 17:04, 21 September 2017 (UTC)[reply]

I've converted to the center dots. I hope I've gotten them all. --Bill Cherowitzo (talk) 04:56, 23 September 2017 (UTC)[reply]

Thank you! 67.198.37.16 (talk) 07:01, 5 December 2023 (UTC)[reply]

There is a move discussion in progress on Talk:Permutation representation (disambiguation) which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 20:31, 15 November 2017 (UTC)[reply]

The transitivity line of the section on types of group actions claims that the special linear group is transitive over a vector space. Surely this need not be true if the vector space has dimension 1 and the field is non-trivial. The special linear group of the real line is just multiplication by 1, which is not a transitive action on the reals. — Preceding unsigned comment added by 140.194.194.252 (talk) 14:25, 22 June 2018 (UTC)[reply]

I replaced the article's statement of the orbit-stabilizer theorem by a more straightforward statement, and was reverted by Jean Raimbault with the edit summary "though the current version is a bit too abstract this one is not clearer". I disagree. I understood the statement which I added at first reading. The current statement requires the reader to look up "image" and coimage; and I have still not understood it after at least six readings. The current version is also followed by mention of "G-sets", while providing no clue about what a G-set is. I accept that the current version may be preferred by readers who already know and understand the orbit-stabilizer theorem, but it is little use to readers who have come to the article hoping to find out. Maproom (talk) 13:47, 25 November 2018 (UTC)[reply]

I agreed that the version before yours was too abstract, and just changed it (with a precise reference to a standard text in algebra).

Your version was in my opinion worse : the citation comes from some notes on a specific problem and the statement that you reproduced is specific to the situation there, and concerns the induced action of the group on the set of subsets of X, not the action on the set X itself. In my opinion this elads to it being easily misinterpreted as that fact was not made clear. jraimbau (talk) 13:54, 25 November 2018 (UTC)[reply]

I was wondering, if we can add the following definition to the second list in the Types of actions section:

• Covering space action if every point ${\displaystyle x\in X}$ has a neighbourhood ${\displaystyle U}$ such that ${\displaystyle \{g\in G:g\cdot U\cap U\neq \emptyset \}=e}$.

Since in the book Algebraic topology Allen Hatcher writes

"This is not standard terminology, but there does not seem to be a universally accepted name for [it]."

I assume this might be a matter of debate whether to list this as a standard definition. But since this book was (and still is) quite influential in the field of algebraic topology I'm asking if it justifies to just do it. K!r!93 (talk) 20:54, 22 April 2019 (UTC)[reply]

I've read the paragraph about the orbit-stabiliser theorem many times now and I just can't understand it. The text is too condensed - it seems to try to introduce the theorem and prove it in a single sentence, and it does so in a way that doesn't build on what comes before it in the article. (This same sentence introduces the notion of coset for the first time, and also uses the word "translate" in a sense that's not familiar to me.)

It seems like this is a pretty important theorem, and the article would greatly benefit from spending a bit more time introducing the concepts behind the theorem and explaining what it shows. Nathaniel Virgo (talk) 19:32, 12 June 2019 (UTC)[reply]

I added some explanation and a very simplistic example. If you are fine with it now, you can remove the tag.(I am not going to explain coset, there is a link to the corresponding page.)ChristianTS (talk) 15:52, 1 October 2019 (UTC)[reply]

"In mathematics, a group action is a formal way of interpreting the manner in which the elements of a group correspond to transformations of some space in a way that preserves the structure of that space."

From this we might conclude:

1. There is some space.
2. It has a structure (whatever that means).
3. There can be transformations of that space.
4. ... which might or might not preserve that structure.
5. There is a group.
6. An element of that group corresponds to (one or some of?) the transformations mentioned in (3, but only if preserving structure (4).
7. There is a manner of 6.
8. There are interpretations of the manner in 7
9. Among the interpretations of 8, some are formal.
10. A group action is one of the formal interpretations of (9).

In short, this is a word salad with so many indirections and hedges that it is either incomprehensible or incorrect. For starters, I am pretty sure that a "group action" is not a "way of interpreting". There may be a "way of interpreting" that allows us to perceive "group action" as a such-and-such, but the group action is not the way-of-interpreting per se.

Perhaps this was the intended meaning:

"In mathematics, a group action is a transformation of a space to a new space that has particular structural properties in common with the first space."

Gwideman (talk) 23:57, 25 November 2019 (UTC)[reply]

While I agree that the introduction of this article is not easy to understand, I would object to your proposed formulation, "In mathematics, a group action is a transformation of a space to a new space that has particular structural properties in common with the first space." Two points:

• Every group element gives a transformation of the base space to itself, so both spaces you wrote about are the same.
• A group action is not a single transformation but a family of transformations, parameterized by elements of some group, with the parameterization subject to the identity and the composability requirements.

(That being said, I am unhappy with the phrasing in the article, "in a way that preserves the structure of that space". To me this seems to imply that group action needs to be faithful, but here I'm probably reading to much into that sentence.)

Anyway, it obviously is quite hard to capture the idea of a group action in prose. – Tea2min (talk) 09:56, 26 November 2019 (UTC)[reply]

I agree that the first sentence must be changed and that Gwideman's suggestion is not better. However, the whole lead is too long and confusing. So, fixing the first sentence needs to completely rewrite the whole lead (and probably the whole article). Nevertheless, I suggest the following for the first paragraph: In mathematics, a group action on a space or a mathematical structure is a group homomorphism of the group into the transformations of the space or automorphisms of the structure. One says that the group acts on the space or structure. If a group acts on a structure, it acts also on everything that is built on the structure. For example, the group of Euclidean isometries acts on Euclidean space, and also on the figures drawn in it, in particular, it acts on the set of all triangles. D.Lazard (talk) 10:54, 26 November 2019 (UTC)[reply]

I reworded the first sentence on 14:52, August 27, 2017 with this edit in an attempt to improve it after another editor complained about the previous wording (see Utterly unhelpful introductory sentence, above). I agree that a definition of a group action is difficult to capture in one sentence in such a way that is accurate, complete, and easy for a layman to understand. I like the introduction proposed by D.Lazard above.—Anita5192 (talk) 18:27, 26 November 2019 (UTC)[reply]

I have been bold and implemented my proposed introduction, before remarking Anita5192's edit. I Have also rewritten the paragraph on permutations. I'll continue by removing non-encyclopedic content that has no real content, or does not belongs to this lead. D.Lazard (talk) 18:48, 26 November 2019 (UTC)[reply]

First, thanks for paying some attention to this. The new formulation of the intro paragraph now suffers from having two only-partially-parallel ideas muddled into the lead sentence, after which it's difficult to tell which of these ideas subsequent sentences and clauses refer back to.

In the following, I have pulled apart these two separate lines of discussion into simpler sentences in an effort to reveal what refers back to what. I noted some ambiguities in [brackets].

1. In mathematics, a group action can act on either a space or a mathematical structure.

2. A group action on a space is a group homomorphism of the group into transformations of the space. [What does "of the group" refer back to here?]

3. A group action on a mathematical structure is a group homomorphism of the group into automorphisms of the structure. In this case the group action also acts on everything that is built with that structure. [Again, what does "of the group" refer back to?]

4. For example the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles. [So this is an example of group action on a space, right?]

5. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. [Again an example of group action on a space, right?]

6. [So no example of group action on a mathematical structure?]

Gwideman (talk) 01:38, 29 November 2019 (UTC)[reply]

I'll take a shot.

A group action of a group on a set is a collection of transformations from the set to itself, such that the collection of transformations under function composition is compatible with, or commutes with multiplication in the group. Similarly, a group action on a mathematical structure is a group homomorphism of the group into the automorphism group of the structure.

There. Perhaps that is less garbled? 67.198.37.16 (talk) 07:57, 5 December 2023 (UTC)[reply]

The new version of the first sentence is slightly better than the preceding one, but it remains overly technical for this place in the article, since the meaning of the definition can only be understood by people knowing already the subject. Instead, the lead must begin with an explanation of the need of the concept. Something like

In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists in performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group acts also on triangles by transforming triangles into triangles. More technically ...

I do not imagine a better way for fixing the issues raised by the opening post of this thread. Nevertheless, my prose may certainly be improved. D.Lazard (talk) 17:43, 5 December 2023 (UTC)[reply]

The part in green sounds entirely fine to me. It's late at night, and all of my brain cells have curled up under the covers and are waiting to go to sleep. So please do copy that in. My only suggesting is that the More precisely ... bit start a new paragraph, so that it's not a big blob of text. I like More precisely ... rather than More technically... since the goal is increased precision, rather than increased complexity. I'll check back later to see if something more eloquent pops into my mind. 67.198.37.16 (talk) 05:10, 6 December 2023 (UTC)[reply]

To also address the comment immediately below, I'm thinking that perhaps a section of 2 or 3 or 4 very basic informal examples be provided, before the definition. Oddly, the intuitvely simplest example I can think of is mathematically complicated, but it works: the rotation group on 3D objects. The canonical imagine flipping a cell phone over, and then turning it to the right; just don't say "non-commutative" and we'll be OK. This can be done with all-words, no formulas. Then perhaps rotations in a plane, because some relatively simple formulas can be given for that, and finally, the simplest/hardest to explain, the permutation group on three objects. Its quite abstract, can lead to confusion, but also allows some notational gymnastics, right before kicking into the formal definition. I'm unlikely to write any of this, unless bitten by some wild hare. 67.198.37.16 (talk) 05:27, 6 December 2023 (UTC)[reply]

I have implemented my above suggestion, and cleaned accordingly the remainder of the lead. So, this section can be closed, and, if there is some problems with this new version, this must be discussed in a new section. Similarly the discussion on the remainder of the article must also occur in new sections. In this long section, there are some posts of this sort. I leave to their authors to repost them in a specific section. D.Lazard (talk) 20:47, 11 December 2023 (UTC)[reply]

Is this change correct? From this text:

Left group action

If G is a group with identity element e, and X is a set, then a (left) group action α of G on X is a function

${\displaystyle \alpha \colon G\times X\to X,}$

that satisfies the following two axioms:

Identity:	${\displaystyle \alpha (e,x)=x}$
Compatibility:	${\displaystyle \alpha \left(g,\alpha \left(h,x\right)\right)=\alpha \left(gh,x\right)}$

(with α(g, x) often shortened to gx or g ⋅ x when the action being considered is clear from context):

Identity:	${\displaystyle e\cdot x=x}$
Compatibility:	${\displaystyle g\cdot (h\cdot x)=(gh)\cdot x}$

for all g and h in G and all x in X.

The group G is said to act on X (from the left). A set X together with an action of G is called a (left) G-set.

From these two axioms, it follows that for any fixed g in G, the function from X to itself which maps x to g ⋅ x is a bijection, with inverse bijection the corresponding map for g⁻¹. Therefore, one may equivalently define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to itself.>

to

Left group action

If G is a group ${\displaystyle \langle X,\cdot ,e,{}^{-1}\rangle }$ with identity element e, and X is a set, then a (left) group action α of G on X is a function

${\displaystyle \alpha \colon G\times X\to X,}$

that satisfies the following two axioms:

Identity:	${\displaystyle \alpha (e,x)=x}$
Compatibility:	${\displaystyle \alpha \left(g,\alpha \left(h,x\right)\right)=\alpha \left(g\circ h,x\right)}$

(with α(g, x) often shortened to gx or g ⋅ x when the action being considered is clear from context):

Identity:	${\displaystyle e\cdot x=x}$
Compatibility:	${\displaystyle g\cdot (h\cdot x)=(g\circ h)\cdot x}$

for all g and h in G and all x in X.

The group G is said to act on X (from the left). A set X together with an action of G is called a (left) G-set.

From these two axioms, it follows that for any fixed g in G, the function from X to itself which maps x to g ⋅ x is a bijection, with inverse bijection the corresponding map for g⁻¹. Therefore, one may equivalently define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to itself.

— Preceding unsigned comment added by 2806:106e:b:c6e9:174:9fc8:e165:a58f (talk) 08:03, 27 December 2021 (UTC)[reply]

I thought that rewriting it with a clearer notation I could understand it, but I failed. Here some questions:

What does it mean to multiply a group, which is a structure, by a set?

Said in other way: What is an action?

Is the "⋅" in g ⋅ x the group operation, or what is it?

Please rewrite it in a more readable way.

I tried but I'm not mathematician.

— Preceding unsigned comment added by 2806:106e:b:c6e9:174:9fc8:e165:a58f (talk) 08:03, 27 December 2021 (UTC)[reply]

For a group G operating on a space X, the dot is commonly used to denote both the binary operation of the group,

⋅ : G × G → G,

and also the action of elements of the group G on elements of the space X,

⋅ : G × X → X,

and the context makes clear which is meant. What's more, both the group operation and the group action are commonly written just using juxtaposition, without any symbol to denote the given operation. I see why this can be confusing on a first reading. The notation is however common (and perfectly standard). (Think matrixes operating on vectors.)

— Tea2min (talk) 06:35, 28 December 2021 (UTC)[reply]

I changed the article to say this, which is perhaps more clear?

It can be notationally convenient to curry the action ${\displaystyle \alpha }$, so that, instead, one has a collection of transformations ${\displaystyle \alpha _{g}:X\to X}$, with one transformation ${\displaystyle \alpha _{g}}$ for each group element ${\displaystyle g\in G}$. The identity and compatibility relations then read

${\displaystyle \alpha _{e}(x)=x}$

and

${\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha _{g}\circ \alpha _{h})(x)=\alpha _{gh}(x)}$

with ${\displaystyle \circ }$ being function composition. The second axiom then states that the function composition is compatible with the group multiplication; they form a commutative diagram. This axiom can be shortened even further, and written as ${\displaystyle \alpha _{g}\circ \alpha _{h}=\alpha _{gh}}$

With the above understanding, it is very common to avoid writing ${\displaystyle \alpha }$ entirely, and to replace it with either a dot, or with nothing at all. Thus, α(g, x) can be shortened to g ⋅ x or gx, especially when the action is clear from context. The axioms are then

${\displaystyle e\cdot x=x}$

${\displaystyle g\cdot (h\cdot x)=(gh)\cdot x}$

Perhaps this removes the confusion of what this supposed to be all about? 67.198.37.16 (talk) 08:36, 5 December 2023 (UTC)[reply]