Talk:Affine group

Mismatched conventions in article

The convention taken in the first part of the article writes the semidirect product with V first and GL(V) second, but later in the "Matrix Representation" section, the ordering is reversed: the elements of GL(V) are in the first components and V in the second components. I would change it myself, but I'm not sure at a glance if the following matrix representations are tied to that particular ordering.

Dzackgarza (talk) 02:11, 14 February 2021 (UTC)

Relation with group theory

I think some link to Frobenius_group and/or Group_theory could be useful. --Pavel Horal (talk) 14:43, 17 January 2008 (UTC)

Typesetting Styles

I didn't realise that there was certain method of typesetting expressions. Thanks for fixing my mistakes. Everything in LaTeX is emphasised in formulae, and so I followed that preconceived typesetting style. Can anyone explain the dos and donts when it comes to numerals and letters? It seems that numerals should not be in italic where as the letters should. Dharma6662000 (talk) 23:24, 16 August 2008 (UTC)

See Wikipedia:Manual of Style (mathematics). In non-TeX mathematical notation, one tries to be consistent with TeX style. That means variables are italicized and digits and punctuation are not. Nor are "det", "max", "lim", etc. And a space preceeds and follows "+", "=", etc. Thus:

5+3=8 (wrong)

5 + 3 = 8 (right)

Also, a minus sign is longer than a mere hyphen:

5 - 3 (wrong)

5 − 3 (right)

I make the spaces before and after binary operators like "+" and "−" non-breakable.

In TeX the software takes care of stuff like this (although you have to write \det, \max, \lim, etc. with a backslash so that it knows it's not just three juxtaposed variables "d", "e", "t", etc.). Michael Hardy (talk) 23:57, 16 August 2008 (UTC)

Oh, okay, I see. I know how to get an unbreakable space in LaTeX: you use a tilde e.g. Theorem~2. How does one do the same in Wikipedia whist outside of maths mode? Dharma6662000 (talk) 00:03, 17 August 2008 (UTC)

Phrasing-"recall that if one fixes a point" ?

If one is just starting on "Affine Group" (s?) then there is nothing to recall. Perhaps a derivation or reference is in order? Ray Rrogers314 (talk) 15:50, 29 February 2012 (UTC)

Quaternions do not form a field

The article currently states "It is a Lie group if K is the real or complex field or quaternions." but K is supposed to be a field according to the first line. Quaternions do not commute and therefore are not a field, right?

I do not know if this means that the first line is wrong or the second line is wrong so I have not made any changes to the article. — Preceding unsigned comment added by 146.186.246.32 (talk) 19:24, 14 October 2013 (UTC)

Planar Affine Group section

The article as it presently stands is misleading as to the nature of the transformations considered in Case 1 of the quoted material from Artzy's text. Looking at the coefficients a and b, and noting that the sum of their squares equal 1, it's clear that this transformation is in fact a rotation. That is to say, it is a similarity transformation, but a special kind of similarity transformation (a rotation! - note, any necessary dilation transformation has already been performed before our three cases begin, as the article correctly observes). So it's misleading to say these are similarility transformations, and leave on the table the additional information that they are in fact rotations.

Therefore I encourage someone to make the necessary edit here - I fear that if I do it, the edit will immediately get reversed for some reason I don't have enough Wikipedia experience or knowhow to counter. I recommend replacing "similarity transformations" to "rotations" and then linking to the Orthogonal Group page. Thank you. — Preceding unsigned comment added by 71.7.135.63 (talk) 17:28, 27 April 2020 (UTC)

The quotation says "followed by a dilation from the origin". This is thus a general similarity. By the way the section is a mess and should be rewritten. One of the issues is that it does not speak of the affine group, but of GL(2, R). That is, it considers only the affine transformations that have a fixed point, and omit the translations. Moreover, the description of the classification of the transformation is unclear and confusing, probably because it is restricted to an out-of-context quotation. I'll tag the section as confusing. D.Lazard (talk) 18:11, 27 April 2020 (UTC)

I have completely rewritten the section. By the way, I remarked that the former version was wrong as excluding the reflections, and other transformations reversing the orientation. D.Lazard (talk) 12:06, 22 February 2021 (UTC)

Are the affine groups parallel or analogue to holomorphs?

Aff(V) is the semidirect product of (V,+) and Aut_K(V), where Aut_K means the automorphism group of V as a K-vector space. This reminds me of the definition of holomorph of a group G which is the semidirect product of G and Aut(G). 2A04:CEC0:C010:1137:18BA:2892:6668:50E (talk) 15:53, 24 November 2023 (UTC)

Holomorphs are generalizations of affine groups: an affine group is the group of affine transformations of an affine space. It is the semidirect product of the group of translations (which you denote (V,+)) and the group Aut_K(V)= GL_K(V), where V is the vector space of the translations. This semidirect-product property is not the definition, it is a property. Geometrically, the action of the affine group on a shape deforms this shape but keeps its appearance (for example, a rectangle is transformed into a parallelogram). So, "holomorphism", which means "similar form" is an old name for an affine transformation. It is presently used for the generalization of the above property, but no more in use for affine groups. D.Lazard (talk) 17:31, 24 November 2023 (UTC)