Talk:Semidirect product

message: Is it intentional that the phi symbol is changed to the varphi symbol in the "Outer Semi-direct product" section?  — Preceding unsigned comment added by 92.237.206.188 (talk) 19:56, 8 February 2019 (UTC)
thread_title:  Confusion of symbols 

message: Look at the section regarding the fundamental group of the klein bottle. it is also the direct product.  —Preceding unsigned comment added by 128.186.24.115 (talk) 13:59, 29 September 2008 (UTC)
thread_title:  Semidirect is direct product for commutative groups 

message: The Unicode standard defines ⋉ (U+022C9) as "[LEFT NORMAL FACTOR SEMIDIRECT PRODUCT]". Mathematical conventions may vary, but it seems best to use the character agreed upon by the international body, and write N⋉H rather than N⋊H. I have replaced three instances, one of which was an inline image of the character. KSmrq 14:56, 9 Jun 2005 (UTC)
replies: text: This symbol is not displayed correctly for me (Firefox/linux). Anyone know of a font with the character? Lupin 18:07, 22 Jun 2005 (UTC)
replies: text: If that one is not displaying, chances are many others are missing as well. A good place to start for Gecko/Mozilla/Firefox browsers is the font page of the MathML project. Also, we recently have excellent news, that the complete repertoire of STIX fonts will finally be released to the public in a few months. For testing fonts, it helps to have a site that catalogs Unicode. KSmrq 06:05, 24 Jun 2005 (UTC)
thread_title:  Symbol 

message: The following statements are equivalent is Bourbaki style. I think it is out of place on WP. Semidirect products are fundamental, and we need a gentler introduction. Equivalent characterisations are things to put later in an article. Charles Matthews 13:05, 20 October 2005 (UTC)
thread_title: Bourbaki talk

message: Wouldn't it be a little more natural to say this group is a semidirect product of Z and Z_2? Since that's what the thing looks like.  I mean, it looks true and all, I've just never seen it as it is here.  It's a little like describing the fundamental group of RP^2\times S^1 as <a,b|a^6b^{-4}>, which is true, but doesn't help most people reading it for the first time.  —Preceding unsigned comment added by 24.59.105.30 (talk) 03:14, 22 October 2007 (UTC)
replies: text: Where did you get the idea that it's a semidirect product of Z and Z_2? The only nontrivial semidirect product is \langle a,\;b \mid b^2=1\; bab=a^{-1}\;\rangle, whose abelianisation is (Z_2)2, so it can't be right. Algebraist 21:33, 29 May 2008 (UTC)

text: I don't know about the topology, but <a,b:aba=b> and <a,b:aba=b,b^2=1> are not isomorphic.  The first is a semidirect product of Z with Z (with <a> normal), and the second is Z with Z/2Z.  The abelianization of the first is Z x Z/2Z and the second is Z/2Z x Z/2Z.  The article switches the roles of a,b versus this talk section, but claims the fundamental group is the semidirect product I gave first. JackSchmidt (talk) 17:51, 30 May 2008 (UTC)
replies: text: The article is correct (by the Seifert–van Kampen theorem). I was just convincing myself that the anon's suggestion differs from that in the article. Algebraist 21:01, 30 May 2008 (UTC)
replies: text: Sounds good to me.  As far as I know (which is the problem, of course) it is basically trivial to calculate the fundamental groups of compact real surfaces.  I think abab^-1 labels the little square diagram of a Klein bottle so it basically sounds right to me. I actually never looked at surfaces whose little diagram thingy wasn't a square.  Are there any (presentations of) fundamental groups of surfaces with two generators that require more than a single relator?  If I had a Klein bottle and stuck a handle on it, doesn't something simple happen to the presentation, like, you add one relator and one generator? HNN extension maybe (which is hopefully a semidirect product)?  I've been pretending to give topological methods a chance, so I probably ought to know this basic stuff.  If it is interesting to you, but dull for this page feel free to reply on my user talk.  If it is dull to you too, then I'll get around to rereading Rotman's chapter on HNN extensions at some point. JackSchmidt (talk) 02:02, 31 May 2008 (UTC)
thread_title:  Fundamental Group of the Klein Bottle 

message: The semi-direct product need not be between two subgroups of the same group. If you have one group, say G, that acts on another group, say H, then one may construct the semi-direct product G \ltimes H.
One important example is given by G := \mbox{Mat}(n,\mathbb{R}) with the operation of matrix multiplication and H := \mathbb{R}^n with the operation of vector addition. Clearly G acts on H in the natural way, i.e. identify x \in \mathbb{R}^n with a row vector and then the action of M on x is just multiplication.
So G gives linear transformations and H gives translations. Then we have \mbox{Mat}(n,\mathbb{R}) \ltimes \mathbb{R}^n with the structure
: (M,x) \cdot (N,y) = (x + My, MN) 
This reflects the fact that applying a linear transformation will alter any translation that went before. Dharma6662000 (talk) 19:16, 30 July 2008 (UTC)
replies: text: This construction is covered in the article, starting with the sentence Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism φ : H → Aut(N), the new group N\rtimes_{\varphi}H (or simply N ×φ H) is called the semidirect product of N and H with respect to φ, defined as follows. Do you think it should be explained differently? Algebraist 23:47, 21 August 2008 (UTC)
thread_title:  Too Specific 

message: Can someone show me the fomula of simidirect product of Lie algebras g->der(h). I only find for h a vector space(Abelian Lie algebra) in books.
 Is [(g_1,h_1),(g_2,h_2)]=([g_1,g_2],g_1h_2-g_2h_1+[h_1,h_2]) ??
--刻意(Kèyì) 16:57, 14 November 2009 (UTC)
replies: text: The Lie algebra and bracket can be found in Lie algebra extension#Semidirect product (groups). YohanN7 (talk) 21:46, 17 May 2015 (UTC)
thread_title: Lie algebras

message: I dont think the inverse element is right.  —Preceding unsigned comment added by 131.174.17.85 (talk) 14:35, 26 October 2010 (UTC)
thread_title:  Inverse 

message: The current statement is that the order of G=N\rtimes H equals the product of the orders of N and H because G is isomorphic to the outer semidirect product of N and H. This is false; I think what was meant was that this is because G has the same order as the outer direct product. Adam Marsh (talk) 17:55, 21 March 2018 (UTC)
thread_title:  Properties 

message: The following sentence should be definitively explained, since the page on fibred categories does not even mention semidirect products.
There are also far-reaching generalisations in category theory. They show how to construct fibred categories from indexed categories. This is an abstract form of the outer semidirect product construction.
151.78.197.39 (talk) 17:22, 30 September 2014 (UTC)
thread_title:  Generaliztions in categories