Thanks! I didn’t know that this is on the arXiv. Also that retyped pdf was uploaded many years ago, I only copied it over here now form *Erlangen program*. In any case, thanks for giving the arXiv link, that indeed makes the other pdf redundant.

Added a link to the arXiv version of Klein’s paper.

Do we really need to have an arXiv paper uploaded also to the nLab as a PDF file?

]]>On a related note: Is the term “G-space” due to Bredon 72? It’s not yet in Koszul 65, as far as I can see.

]]>added also pointer to:

- Jean Louis Koszul,
*Lectures on Groups of Transformations*, Tata Institute 1965 (pdf, KoszulGroupsOfTransformations.pdf:file)

on the usage of the term “transformation group” for “group action” I have added pointer to Section 1 of:

Felix Klein,

*Vergleichende Betrachtungen über neuere geometrische Forschungen*(1872) Mathematische Annalen volume 43, pages 63–100 1893 (doi:10.1007/BF01446615)English translation by M. W. Haskell:

*A comparative review of recent researches in geometry*, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (euclid:1183407629 KleinRetyped.pdf:file)

22, 23, 24 While I strongly agree with sensitive discussion in 23, I still think that the summary in 24 is missing parts of the point from the mathematical practice, even though major authorities quoted in 22 have overdefined in this vain. A modern practical mathematician will indeed tend to call transformation group a group with some **standard** (or often even defining) action of it, for example the diffeomorphism group of a manifold, or permutation group on $n$ letters. On the other hand, when one is fixing a group and considering varying actions, “realizations”, faithful or not, nowdays one would like to call it as action or realizations, or (nonlinear) representations, rather than varying transformation groups.

25 While doing group theory abstractly without their actions is loosing the real power of the theory (as Miles Reid used to criticise), the reducing point of view which I heard from a professor as a junior student that a group is interesting only when it is realized as a linear representation is also at lest demotivating (for me, that blatantly reductionistic point of view (“group makes sense only if realized as matrices”) has pushed me away psychologically from representation theory for my entire career). For example, combinatorial group theory with free groups and their subgroups has its own content which start with combinatorics of generators and relations, and then introduces connections to topology, automorphisms etc. (mainly non-linear). I knew the beauty of combinatorial group theory from my hi school having read Magus’s children book on groups and their graphs, and then as a freshman first pages of deep book by Lindon and Schupp. Then martyrs of the law nothing makes sense if it is not a concrete matrix killed my love for group theory for many years. Mathematics has many facets and insisting that only one point of view is right is doing more harm than good. Of course, the questions in 25-29 are interesting.

]]>Hmm, actually it makes a big difference to require to not have any representations in dimension higher than 3, this will exclude the simple groups I think, and maybe the naive guess (abelian normal subgroup of index 2 or 3) will work. Maybe this does generalise to all dimensions (i.e. one asks for abelian normal subgroups of indices corresponding to the prime decomposition).

]]>Actually I think that finite groups with only 1- or 2-dimensional irreducible complex representations are exactly what one would naively guess: those groups with an abelian normal subgroup of index 2. So everything works out quite nicely up to dimension 2. I’m not sure if one will be able to say anything in dimension 3 though.

]]>For something easier, one can recognise abelian groups by their representations: they are exactly those groups with *only* 1-dimensional (irreducible) representations. Thus one does kind of see simpleness roughly reflected in the 1-dimensional representation theory: maximally non-simple groups (i.e. abelian groups) have only 1-dimensional representations, maximally simple groups have none.

Simple groups have no 2-dimensional representations either, but I’m not sure if groups with *only* 2-dimensional representations are interesting.

(Certain simple groups have a 3-dimensional representation.)

]]>It would be interesting to try to study groups *only* via their representations. Take simple groups for example. Can one recognise a simple group through its representation theory? They have no 1-dimensional representations over $\mathbb{C}$, but that is not a characterisation. However, can one for example say anything interesting about those groups with no 1-dimensional complex representations as a class?

Group actions… are also known as

transformation groups…

Speaking of a situation

$\alpha: G \times X \to X,$I would call $\alpha$ the group action, and $G$ the transformation group.

You can see how the shift to plain “group” took place as one looked to study such $G$ without reference to any specific $X$.

]]>Maybe there is a historical comment in order, as these days “transformation groups” seems not be used at all anymore?

As an aside, I find there is a missed opportunity here in fixing up the terminology:

After all, “group” is a somewhat unfortunate abbreviation of “group of symmetries” (unfortunate because it dropped the essential meaningful aspect while keeping an *ad hoc* technical term) and that in turn originates in “groups of symmetry transformations”, as originally (i.e. historically) all “groups” were really understood through/as their group actions, in the first place. So in speaking of “transformation groups” one is going back to the roots of the subject while insisting on historically evolved twists of what would have been its natural terminology.

Imagine in a bookshelf seeing the title “Symmetry transformations” – everyone knows at once what it’s about. Instead we have “Group theory” which (if you manage to remember what this looks like to the non-expert eye) is so unevocative of its subject matter that even modern search engines keep mixing it up with “Group therapy” – and that’s well-deserved, if overly mild, punishment for being so careless about the naming of concepts so profound.

]]>That’s a good point to highlight. I have expanded this as follows:

Group actions, especially continuous actions on topological spaces, are also known as

transformation groups(e.g. Bredon 72, tom Dieck 79, tom Dieck 87). Alternatively, if the group $G$ that acts is understood, one calls (Bredon 72, Ch. II) the space $X$ equipped with an action by $G$ atopological G-space(or G-set, G-manifold, etc., as the case may be).

and added pointer to these references.

]]>Redirect: transformation group.

Added:

Group actions, especially on spaces, are also known as *transformation groups*.

made explicit the action property for group actions (here)

$\underset{ g_1, g_2, s }{\forall} \;\;\; \rho(g_1 \cdot g_2)(s) \;=\; \rho(g_1) \big( \rho(g_2)(s) \big)$so that one can refer to it.

]]>Fixed a dead link to a pdf file

]]>Added link to category MSet.

]]>Tried to make the point about profinite groups not necessarily having internal automorphism objects more clear, and gave a simpler example in addition (topological groups acting on topological spaces),

]]>Adding section with link to page-to-be-created on actions of a profinite group. Surprisingly difficult to find a good reference when googling.

There is a paragraph just before the new section which may later be incorporated into the new page or otherwise edited.

]]>Added clarification to how left and right actions are mirrored in variance of the functor.

]]>David: yes, it seems that way.

]]>(Small terminological comment on

Is this somehow related to a category of quivers with edges labeled by $L$ is the slice category $Quiv/Rose(L)$ where $Rose(L)$ is the one object quiver with $L$ edges?

from 9: in my opinion, the only sensible term for “the one object quiver with $L$ edges” is

*one-vertex quiver with $L$ edges*

which is also attested here and there on the web. Using “one object” seems misleading, or at least wrong emphasis, to me, for known reasons: a quiver is not a category; each quiver consists of sets, and calling sets objects is not wrong but can be wrong emphasis. )

]]>Re

Is “$Rose(X)$” standard notation in quiver-theory?

from 10: not to my knowledge. Absence is hard to prove, yet I think there simply does not exist any usual term for this. In *undirected contexts*, people talk about *bouquets* a lot (flowery that, too), but in the directed setting: no, I think not.

Would there be discrete fibration-like lifting conditions on the map of quivers? The definition doesn’t require the composition operation.

]]>