# Talk:Symmetry group

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Field:  Geometry

## 1

My 2c on this page - it has useful info but could be made more readable and more concrete. Symmetry groups are a great starting point for finding out about finite group theory -- especially the symmetry groups of plane figures and archimedian solids, but this article dives in with "congruencies", "invariant" and "composition". Most non-mathematical readers (including proto-mathematicians) would not survive the first sentence. The link to Group theory doesn't help much either. ... and subheadings would help.

If I get around to it in the next few weeks I will put in a gentler introduction and some examples that will ease the transition into the language of the two and three dimensions sections. I am not a groupie though, so if anyone else has the skills and energy, I will not be offended. AndrewKepert 06:15, 12 Nov 2003 (UTC)

How about, "The symmetry group of a geometric figure is the set of symmetry operations, such as rotations and reflections, which leave the figure indistinguishable from its original form. For example, a square can be rotated a quarter of a turn, and it is still the same square. It can be also reflected about lines through the center parallel to the edges or passing through opposite corners." --Snags 20:39, 8 Oct 2004 (UTC)

"Td. This group has the same rotation axes as T, but with six mirror planes, each containing a single C2 axis and four C3 axes." You mean two C3 axes, right? A plane cannot contain all four C3 axes.

## Request for technical explanation

Probably the best thing to do would be to include a concrete example with diagram(s) in the introduction. (Perhaps what Snags suggests above.) The current introduction gets into a lot of details which are hard to parse, and which should probably be put into subsections. The 1D, 2D, and 3D sections could certainly also use some diagrams or pictures showing concrete examples. -- Beland 17:00, 18 December 2005 (UTC)

## editorial templates formerly in article text

(This page currently does not yet describe various aspects of symmetry groups in theoretical physics, especially in (quantum and classical) field theory.)
• The article has had a link to a DAB page which lists the physics symmetry groups for many months now. Debivort 21:00, 17 January 2007 (UTC)

## the case of the comma

The last editor apparently read a book that said i.e. (id est = 'that is') and e.g. (exempli gratia = 'for example') are always followed by a comma. Bad book. If they are followed by a comma, they ought also to be preceded by one; see Parenthesis (rhetoric). But these additions brought my attention to some passages that could be improved, so thanks all the same!

Sometimes a broader concept of "same symmetry type" is used, resulting in[,] e.g., 17 wallpaper groups.

I removed this sentence because I can't make sense of it. Does it mean that wallpaper group is itself a broad symmetry type, and thus all 17 are in a broad sense "the same"? Or that, with a narrower understanding of "same", there would be more than 17 of them? —Tamfang (talk) 06:07, 8 December 2008 (UTC)

Actually, "i.e." and "e.g." are parenthetical, which means that they should be punctuated accordingly. Generally this means following them with a comma and preceded with some other form of punctuation which may or may not be a comma. So I do not agree with the conditional proposition in the above post beginning "If..." siℓℓy rabbit (talk) 13:24, 8 December 2008 (UTC)
We agree that they're parenthetical; the previous editor mechanically added a comma without adding any preceding punctuation, as in the line quoted. —Tamfang (talk) 03:42, 12 December 2008 (UTC)

## Merge

I would like to start a discussion to propose the merge o the Symmetric group to this one. Clearly both articles refer to the same concept, despite the fact that the other article states that it focus on "finite sets". --Pedro 20:49, 6 May 2010 (UTC) —Preceding unsigned comment added by Pcgomes (talkcontribs)

I disagree that the two articles refer to the same concept. The symmetric groups are a particular sequence of groups of orders 1, 2, 6, 24, ... . In contrast, all groups are symmetry groups, and symmetry groups exist of all orders. —Mark Dominus (talk) 01:21, 7 May 2010 (UTC)
The symmetric groups are also called permutation groups. They are groups on sets in which the arrangement of any subset is independent of the arrangement of other subets, which is not true of (for example) the vertices of a polyhedron (other than a simplex). —Tamfang (talk) 01:53, 7 May 2010 (UTC)
do not merge - per above. They are separate concepts deserving separate articles. de Bivort 01:57, 7 May 2010 (UTC)
Symmetric groups, permutation groups and symmetry groups are all different concepts. The first notion is absolute (the adjective depends only on the group: a given group with 168 elements (any one) is not a symmetric group, period) but the other two are are relative (the chosen group could be realized as permutation group on some set of elements, or as symmetry group of some configuration, both of which would need to be specified for the statement to make sense). Merging the articles would be a blunder, reinforcing the confusion possibly caused by the terminology. Marc van Leeuwen (talk) 04:51, 7 May 2010 (UTC)

## Major mistakes in section about symmetry groups of two-dimensional objects

In the section Two dimensions one assertion reads as follows:

"D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle . . . "

But this is not true. In fact the entire section is filled with mistakes like this. The symmetry group of a non-equilateral rectangle has 8 elements, not 4. Since every element is of order 2, this must be isomorphic to the group Z/2Z ⊕ Z/2Z ⊕ Z/2Z.

The section discusses only orientation-preserving symmetries, and it does this without saying so. There is no particular reason to restrict to only orientation-preserving isometries, and even though the discussion is about 2-dimensional objects there is no reason for a reader to assume that symmetries need to be orientation-preserving.

On the other hand, orientation-preserving symmetry groups (of orientable objects) are interesting in their own right, so in my opinion they should be mentioned as long as they are clearly labeled as to what they are.

And so this needs to be fixed.Daqu (talk) 16:54, 20 June 2015 (UTC)

replacing my previous edit I think we should find a suitable figure, then. —Quondum 20:45, 20 June 2015 (UTC)
Fixed: non-equilateral rectangle → non-equilateral isosceles triangle. —Quondum 20:49, 20 June 2015 (UTC)
I count only two symmetries for a non-equilateral isosceles triangle: the identity and the median through the different side. Which are the other two? --DonBex (talk) 00:02, 8 January 2016 (UTC)
This fix is wrong, the symmetry group of a non-equilateral isosceles triangle is not isomorphic to the Klein-four group. Edit by 201.50.93.196
I will start with two general statements before answering the specific questions. First, it is important to remember that this section deals with two dimensions. Some of the above statements are valid for three dimensions but not with two. And second, the elements of a group are symmetry operations such as rotations and reflections, rather than symmetry elements such as axes and planes.
In two dimensions, the isosceles triangle has only two symmetry operations = the identity operation and a single reflection in the plane which bisects the triangle. The rectangle has four symmetry operations: the identity operation, reflections in two planes, and rotation by 180o. So the rectangle is isomorphic to the Klein-four group and the isosceles triangle is not. DonBex's original fix was therefore correct in two dimensions and I will restore it. I will also add the list of the symmetry operations.
Note however that in three dimensions, the isosceles triangle has four symmetry operations and the rectangle has eight. In the Schoenflies notation of spectroscopy and chemistry, the isosceles triangle has symmetry group C2v which is isomorphic to the Klein-four group, while the rectangle has symmetry group D2h. These facts may be the source of the confusion, but they are not strictly relevant to the section on two-dimensional objects. Dirac66 (talk) 01:15, 8 January 2016 (UTC)

## Fundamental mistake in introductory sentences

"In abstract algebra, the symmetry group of an object (image, signal, etc.) is the group of all isometries under which the object is invariant with composition as the operation. It is a subgroup of the isometry group of the space concerned."

There is no reason in the world for an object to have a "space concerned".

Also there are many symmetry groups of objects that are not metric spaces, so the symmetry group has nothing to do with preserving distances. It would be a group that preserves some property.

For example: The symmetry group of a set is just all permutations of elements of the set. The symmetry group of a Riemann surface is the group of conformal transformations of the surface. The symmetry group of a topological space is its group of self-homeomorphisms. The symmetry group of a group is its group of automorphisms.

In general, the symmetry group of any mathematical structure is the group of invertible self-mappings that preserve that structure.Daqu (talk) 20:29, 20 June 2015 (UTC)

I disagree that the space has nothing to do with it. The space is the mathematical object affects the size of the group of an embedded object's symmetries.
I agree with your remaining statements. The concept of a symmetry group relates to he preservation of chosen properties, and isometry is specific to a space with a metric. We should remove the isometry-centric assumption. —Quondum 21:02, 20 June 2015 (UTC)
Daqu comes close to what I have seen as "good" definitions in the literature. Permit the structure "no structure", except the set itself (permutation group) and a symmetry group of a structure on a set is a subgroup of the permutation group preserving that structure on that set. I don't think it is necessary to include "images" and "signals" explicitly, they are represented as sets. Can't remember exactly from where I got this. The definition we give must absolutely be referenced. YohanN7 (talk) 20:18, 15 July 2015 (UTC)

## Transformation

Shouldn't the transformations in the symmetry group of for instance a triangle be isometries?Madyno (talk) 13:35, 23 July 2017 (UTC)

I would say it depends on what one means by "triangle". If it is a geometrical triangle, then yes, the symmetry group in mind is likely the isometry. If it is a triangle graph (for example), then no, it would not be because there is no distance structure to preserve. In general, what one means by "symmetry" is always in the context of what structure one is preserving. That may include the distance, or it might not. Sławomir Biały (talk) 16:42, 23 July 2017 (UTC)

Right, yes, but my proble is that not every transformation of a triangle, that maps the triangle on itself is a symmetry. Yet the definition says so. Madyno (talk) 18:42, 23 July 2017 (UTC)

Not knowing the specific context you're referring to, I would guess that by "transformation" is intended Euclidean transformation. In that case, the statement is true, but could perhaps be clarified. Sławomir Biały (talk) 19:23, 23 July 2017 (UTC)

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