Talk:Semidirect product

Confusion of symbols

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)

Semidirect is direct product for commutative groups

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)

Symbol

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)

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)

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)

Simmetry

simmetry there is no simmetry in the def, one group is normal, another is not, so I remove it again. Tosha

The definition is symmetrical in the following sense:

Let N be a normal subgroup of G and H be a subgroup of G. The following are equivalent:

• every element of G can be written in one and only one way as a product nh, with n in N and h in H.
• every element of G can be written in one and only one way as a product hn, with h in H and n in N.

Of course, in both cases, we write G = NXH. Maybe you misunderstood my statement to mean that NXH is the same as HXN? I'm not claiming that, since it is false, and it fact meaningless as you point out. I'll try to clarify. AxelBoldt 21:06, 17 Sep 2004 (UTC)

Sure, I understand, but your statement might create a lot confusion. You say simmetric and reader might guess what ever he wants. On the other hand I do not think that it deserves further explanations. I mean, I do not see any usefull information in this par, but I will not remove it again (hope somebody else will) Tosha

Well, if it helps I can leave out the word "symmetric". I think the useful information for the reader is this: the notation G = NXH seems to imply somehow that the elements of N are multiplied from the left with the elements of H, i.e. that G = NH. And our definitions appear to imply the same thing. So in order to remove this possible misunderstanding, I added that G = HN just as well: there was in fact no inherent reason to have chosen a notation where the N sits on the left of H, it's just a convention.

But I certainly don't want to create a lot of confusion, so if you can think of a way to say the above without confusing the reader, please jump right in. In fact, I believe I just thought of a way, let me know what you think. AxelBoldt 18:45, 18 Sep 2004 (UTC)

What does G = NH mean?

I couldn't find a definition for this notation, neither in Group (mathematics), nor in Direct product, even though it is used there as well. Presumably, it is different from both N⋊H and N×H, since those are used concurrently. — Sebastian (talk) 22:25, July 12, 2005 (UTC)

NH = { nh | n in N and h in H }. --Zundark 22:40, 12 July 2005 (UTC)

Thanks! Is there a name for this? Then we could reference it. — Sebastian (talk) 23:29, July 12, 2005 (UTC)

I am inclined to call these things subset products or subgroup products, but neither is standard (so far as I know). I don't know of any standard phrase that would uniquely identify this concept. - Gauge 21:34, 9 January 2006 (UTC)

Well, we do have an article on this called product of groups, but that is a particularly bad choice of names. Perhaps product of group subsets would be better. -- Fropuff 21:50, 9 January 2006 (UTC)

In Dummit and Foote's algebra text, this is just called N meet H. —The preceding unsigned comment was added by 72.151.105.202 (talk) 00:37, 8 December 2006 (UTC).

If you're only talking about subgroups, I would think this would be the join, $N\vee H$, (not the meet, $N\wedge H$, the intersection is meet) of N and H, see Lattice of subgroups. Summsumm (talk) 10:20, 20 May 2009 (UTC)

Cartesian product

Are Cartesian product and direct product used synonymously here? If so, then the link to the former should be removed. — Sebastian (talk) 22:25, July 12, 2005 (UTC)

No, the direct product of N and H is the group N×H, and the Cartesian product of N and H is the set N×H. --Zundark 22:40, 12 July 2005 (UTC)

Thank you - this explains my confusion in the Frieze group article, where group #7 is described (in the third paragraph) as "Z × {0}", and then as "a semidirect product of Z × C₂ with C₂". I guess the former should be changed to "set", then - or maybe cut altogether. — Sebastian (talk) 23:35, July 12, 2005 (UTC)

That was a (somewhat odd) description of an example pattern, a row of dots, not of the symmetry group. It was removed, "row of dots" is enough.--Patrick 14:09, 5 August 2005 (UTC)

Use of variables not standardized

This article writes G = N×H, while direct product uses K = G×H. How about using the same variables as far as possible, as in K = G×H, K = G×N or K = N×G (if writing the normal subgroup first is conventional or advantageous)? — Sebastian (talk) 22:25, July 12, 2005 (UTC)

Bourbaki talk

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)

Actually I think the outer case ought to come first. That is, treat the semidirect product as a construction, not a recognition problem, initially. Charles Matthews 20:03, 20 October 2005 (UTC)

Semidirect product symbol

(Copied from Wikipedia talk:WikiProject Mathematics#Semidirect product symbol.)

The common notation of a semidirect product seems to be G = N [[Image:Rtimes2.png|]] H, with the normal subgroup at the left, while the symbol is a cross with a vertical bar at the right (see e.g. [1]), although the names of the symbols seem to suggest that the bar should be at the side of the normal subgroup ([2], [3]). Have other people any thoughts?--Patrick 13:37, 20 October 2005 (UTC)

Starting with a presumption of what "the common notation … seems to be" based on one article on one website, PlanetMath, comes across as rather POV. Wouldn't the AMS site be more authoritative? Or a survey of journals, or textbooks? In the spirit of NPOV it may be worth observing that one Unicode technical report, TR #25, in discussing semantics of operators [4], says this (in part):

• The Unicode Standard does not attempt to distinguish all possible semantic values that may be applied to mathematical operators or relational symbols. It is up to the application or user to distinguish such meanings according to the appropriate context. Where information is available about the usage (or usages) of particular symbols, it has been indicated in the character annotations in Chapter 16, Code Charts in The Unicode Standard, Version 4.0 [5] and in the online code charts [6].

Unfortunately, the chart has no such information about the character in question. --KSmrq^T 19:22, 20 October 2005 (UTC)

My training is in algebra, and I've never seen this before. Septentrionalis 20:33, 20 October 2005 (UTC)

The only group theory textbook I have is Rotman's An Introduction to the Theory of Groups. In it he uses the notation K ⋊ Q where K is the normal factor. I believe this to be a fairly authoritative reference. At any rate, it seems to make the most sense to me that the bar should be on the side of the nonnormal factor (so direct products, with both factors normal, have no bars). -- Fropuff 05:57, 21 October 2005 (UTC)

The AMS site linked above doesn't assign any meaning to the symbol ⋊ (U+22CA). It simply gives it the name rtimes (which is also the TeX name). -- Fropuff 06:03, 21 October 2005 (UTC)

Which AMS site are you referring to? The one I gave [7] has a table with these descriptions for "⋉", "⋊", "⋋", and "⋌":

Unicode	AFII	Elsevier name	AMS name	9573-13 name	Unicode description
22C9	EED6	⋉	ltimes	ltimes	left normal factor semidirect product
22CA	EED7	⋊	rtimes	rtimes	right normal factor semidirect product
22CB	EED8	⋋	leftthreetimes	lthree	left semidirect product
22CC	EED9	⋌	rightthreetimes	rthree	right semidirect product

The ltimes and rtimes symbols with the "normal factor" description have been around for a over a decade. I'm extremely uncomfortable using Rotman's book alone as a notation guide. Yes, it's not too old and the content seems good; but it's one book by one author, and a common complaint on Amazon [8] is about its notation (and typographical errors)! I've just seen too many notations that vary from author to author, all quite respectable; Wikipedia itself is full of inconsistent articles. We really need a wider survey, and maybe some guidelines from AMS or Elsevier or a few journals. Something that is quite common is that followers of one school use one notation, passing it from advisor to student to student, or web sites copy mistakes like juicy rumors and urban myths. I don't have a dog in this hunt, because I learned with neither ltimes nor rtimes, only subscripted times. But given the Unicode description, repeated on the AMS site, I need convincing that the normal group is on the open side. (See below) --KSmrq^T 04:00, 23 October 2005 (UTC)

AFAIK the idea behind this symbol is that it combines the relations $H<G$ and $N\triangleleft G$ (meaning "subgroup" and "normal subgroup", resp.), hence the bar is on the side of the non-normal subgroup.--Gwaihir 13:04, 21 October 2005 (UTC)

Thanks, I added some of this to the article.--Patrick 13:27, 21 October 2005 (UTC)

How does this make any sense? In stating that N is a normal subgroup of G, the bar goes with G and the point goes with N. In stating that H is a subgroup of G, the open side goes with G and the point goes with H. I find it unconvincing that these notations should flip to combine as claimed when we take the semidirect product of H and N. (Also, typically the subgroup relation is written H ⊂ G, using the subset symbol.) In fact, a more compelling argument says that the bar goes with the normal group, because when we multiply pairs (n1,h1) and (n2,h2) as described under "Outer semidirect product", the H components multiply the same as for a direct product (no bar), while the N components go through a homomorphism (the bar). Neither explanation seems consistent with the &lthree; and &rthree; symbols! But we know perfectly well that people devise mnemonics for conventions no matter how arbitrary they seem. ( Stars are ordered by color temperature: O, B, A, F, G, K, M; they have a mnemonic.) Handwaving "explanations" like this are not up to decent scholarly review standards; they definitely are not adequate to decide the choice of symbol (either way). It is irresponsible and POV to claim "The common notation is G = N [[Image:Rtimes2.png|]] H, with a cross with a vertical bar at the right", when the matter has not been resolved. --KSmrq^T 04:00, 23 October 2005 (UTC)

I don't see the "flip". It's just $>H$ and $N\triangleleft$ printed in the same space.--Gwaihir 10:46, 23 October 2005 (UTC)

A common notation. WP doesn't resolve matters; where there is doubt, just add 'but this is not universal'. Charles Matthews 07:45, 23 October 2005 (UTC)

Even worse, we have no documentation of how common the notation is. In mathematics this is called a "conjecture", not a fact (theorem). Opinion masquerading as fact has no place in an encyclopedia article. Therefore I have removed all contested symbols from the body of the article, and appended a "Notation" section to describe the options without taking sides. I used UTF-8 characters since they display fine for me, but anyone crippled by a deficient browser or fonts should feel free to replace these four instances with images. --KSmrq^T 00:15, 24 October 2005 (UTC)

Rotman's book on group theory is a standard reference book on the subject. I believe we can take his notation to be common, if not actually standard. But since you still seem to find this objectionable, here are a few other references using this notation (found using some web searches):

• Alperin and Bell (1995), Groups and Representations. p. 20.
• CRC Standard Mathematical Tables and Formulae, 31st Edition, p. 187.
• Dummit and Foote (2003), Abstract Algebra, p. 177.
• Cameron (1999), Permutation Groups, p. 9.

In particular, consider the following quote from Alperin and Bell:

Let G be a group. Suppose that G has a subgroup H and a normal subgroup N such that G = NH and N ∩ H = 1; then we call G the semidirect product of N by H, and we write G = N ⋊ H. (This notation is common, but not standard; other possible notations include N ⋉ H and H ⋊ N, and some authors do not adopt a notation.)

I was unable to find a single mathematics reference using an alternative notation. You have provided no sources except the Unicode character description (not exactly a mathematics source). Again, I submit that we can take Rotman's notation as very common. -- Fropuff 17:59, 24 October 2005 (UTC)

The page on mathematical conventions wasn't really designed for this kind of question. But I suggest that the discussion be taken there (and not closed down too quickly), as an orderly way forward. Charles Matthews 18:06, 24 October 2005 (UTC)

I don't really see this as a big dispute. User:KSmrq seems to be the only one who disagrees with the suggested notation. My request is that we use Rotman's notation in WP with a note to the effect that this notation is not universal or standard in mathematics (similiar to the quoted statement above). If KSmrq still strongly objects to this request we can move the discussion to the mathematical conventions page as you suggest. -- Fropuff 18:12, 24 October 2005 (UTC)

Also "left normal factor semidirect product" may mean "symbol for semidirect product (which involves a normal factor) with the bar on the left" instead of "symbol used when the normal factor in a semidirect product is on the left". Comparing with names like "left bracket", where the symbol itself is on the left, is inconclusive.--Patrick 10:15, 26 October 2005 (UTC)

Although I am puzzled by my inability to find an example of someone using the bar on the side of the normal subgroup, I have seen people be adamant about this notation, so it probably is a big dispute, and there does not seem to be enough evidence here to warrant a standard notation yet. In lieu of such a standard, I recommend what I have seen many authors do: say "the semidirect product given by H acting on N" or similar, or use the H ×_ϕ N notation and mention that the normal subgroup is N (or whatever). - Gauge 22:19, 9 January 2006 (UTC)

I have provided five references above (some of which are standard references for group theory) for authors using the bar on the side of the non-normal subgroup. I was unable to find any references using the opposite notation and nobody else has provided any either. I think that qualifies for sufficient evidence for adopting this notation. Why the resistance? -- Fropuff 22:37, 9 January 2006 (UTC)

As noted above, I have heard other mathematicians insist on the other notation as standard. Also, I feel that a sufficient survey would have to include more than just 5 references. I wonder what group theorists like Michael Aschbacher or Daniel Gorenstein would say? I don't have a copy of Finite Groups by Aschbacher but I would consider that a helpful reference if it has anything on this issue. - Gauge 00:27, 10 January 2006 (UTC)

Who insists on the other notation? Please provide references. Aschbacher uses the obscure S(H,N,φ) for N ×_φ H (with N normal). I don't have access to Gorenstein. Let me point out that notations like N ×_φ H are no less ambiguous than other notations. For example, is the φ written on the normal side or the non-normal side? Or, is the normal side always on the left or the right? We cannot avoid the fact that we need choose one convention and stick with it. Our only guide to doing so is to reference the literature, so please provide references. -- Fropuff 01:01, 10 January 2006 (UTC)

I actually wouldn't mind Aschbacher's notation. I don't think as much confusion would arise from using ×_φ in comparison with one of the other notations, so long as we standardize on the φ always being on the righthand side, regardless of which group is normal. This would force everyone to explicitly identify the normal subgroup and φ, which I would argue is a good thing. Please consider supporting this proposal. My references are from my personal communications, and I am not mentioning names to respect their privacy. You can choose to ignore them; I'm simply pointing out that there is not consensus among mathematicians. - Gauge 17:49, 10 January 2006 (UTC)

I would agree that there isn't a consensus; that much is clear. Stating which subgroup is normal will remove any ambiguity in the notation, so I think that should be a strong recommendation. I just don't think we should prevent ourselves from adopting some notational convention for clarity of exposition (as well as to ensure a consistency across articles). I don't think there is anything wrong with saying something like "The group G can be expressed as a semidirect product K ⋊_φ Q where K is the normal subgroup of G given by ...". We can give readers fair warning on this page that this notation, although common, is not universal. -- Fropuff 23:20, 10 January 2006 (UTC)

That sounds reasonable. As I understand it we can agree upon using K ⋊_φ Q where either K or Q could be normal, the symbol φ should always be present and to the right of ⋊, and the normal subgroup should always be explicitly specified. Additionally, I think the action φ should either be specified or otherwise a reference given in the case that it is too complicated to describe without distracting from the rest of the article. Do you agree? - Gauge 04:45, 11 January 2006 (UTC)

Almost. I am insisting that we need to pick a convention as to which side the bar goes on (if it is used). For example if K is the normal subgroup, acceptable notations would be

K ⋊ Q or Q ⋉ K

but not

K ⋉ Q or Q ⋊ K

I suppose the map φ should always go on the same side as the bar (if it is used). Without a bar any combination would be fine (so long as φ is explicitly written). The map φ should always be mentioned explicitly unless it is clear from context. Mostly this happens if Q is a subgroup of Aut(K) and φ is just inclusion. For example the affine group of a vector space V can be written

Aff(V) = GL(V) ⋉ V

Here the action of GL(V) on V is the obvious one. In all cases the normal subgroup should be mentioned explicitly to avoid confusion. If all this sounds good, I'll make a proposal on Wikipedia:WikiProject Mathematics/Conventions. -- Fropuff 18:09, 11 January 2006 (UTC)

I do not agree that we need to have a convention for which side the bar goes on. Identifying which group is acting and which is normal is the crucial part, and the flexibility in notation between normal group facing the bar and normal group facing no bar would make proponents of each happy. I find all four example notations above acceptable. As above, I prefer that the φ goes on the righthand side regardless of which symbol is used and regardless of which group is normal. Why not use the convention ×_φ where φ always appears on the righthand side regardless of which subgroup is normal, and the normal subgroup may appear on either side? I would be happy with that. - Gauge 20:12, 11 January 2006 (UTC)

(edit conflict) Only using ×_φ prevents someone from writing things like GL(V) ⋉ V where the φ is unnecessary. -- Fropuff 20:43, 11 January 2006 (UTC)

Hmmm. My whole point here is that we should pick a convention and stick with it. Some reasons:

• Mathematics textbooks and articles pick a convention and use it consistently. We should do the same.
• An "anything goes convention" is bound to lead to mass confusion. Even if a given article is self-consistent in its notation, different articles may using a conflicting notation.
• What's the point of using a bar if it won't help distinguish which subgroup is normal?
• Different editors may bicker constantly over their favorite convention, switching back and worth at whim.
• Studies have shown that people rarely read articles word for word. They scan instead. Mathematical notation is one thing that is scanned. People who are familiar with our choice of convention needn't read the text to find out which subgroup is normal; it is clear on sight. (Of course, those who aren't familiar with the notation can check the text to see which group is the normal one).

The disadvantage of picking a convention is that we can't please everybody. This is an unfortunate fact of life and is unavoidable. I realize a choice of convention is somewhat arbitrary. But I'd rather we choose something rather than everything (or nothing at all). -- Fropuff 20:43, 11 January 2006 (UTC)

I have no opinion about which notation is right, but do agree with Fropuff that having a convention is good. -lethe ^talk 22:38, 11 January 2006 (UTC)

I agree that having a convention is a good thing. My position is that having a convention regarding which side the normal subgroup should be on when using a bar notation is likely to lead to conflicts. Notational conventions are advisory rather than mandatory, so a convention using bars may not help. The convention ×_φ, while having the minor disadvantage you pointed out above, has the advantages:

• The normal subgroup is not clear from the notation and must be specified. This provides some extra encouragement to specify the normal subgroup.
• The symbol φ is present, reminding the author to specify the action, even if it seems "obvious" to them.
• The "everything convention" with bar notation comes up because of the conflict of opinion on how the bar notation should be used to identify the normal subgroup. The ×_φ notation can have no such conflict because the φ always appears on the right.

In my opinion, the bar notations cause too much conflict and should not be used at all. I strongly recommend this alternative notation. - Gauge 23:25, 11 January 2006 (UTC)

There are those who will want to use the bar notation and we should have a convention in place. As you say it will be only be advisory. We can strongly suggest that editors use the ×_φ notation in favor of the bar notation to avoid conflict. -- Fropuff 00:06, 12 January 2006 (UTC)

I can agree to that. - Gauge 00:33, 12 January 2006 (UTC)

I've made a proposal at Wikipedia:WikiProject Mathematics/Conventions. Let's hold further discussion there. -- Fropuff 01:35, 12 January 2006 (UTC)

Fundamental Group of the Klein Bottle

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)

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)

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)

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)

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)

Too Specific

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)

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)

It seems to me that exactly this is at work for the Poincare group being the semidirect product of the Lorentz group and the group of translations. In fact, the Poincare group article links to here. Couldn't the Poincare group be included as an important example!?

As another example, the unitary group U(N) should be the semidirect product of SU(N) and U(1) (it also links to here). Could someone verify this and include/prove it here!? — Preceding unsigned comment added by Fazhbr (talk • contribs) 12:07, 15 March 2018 (UTC)

Lie algebras

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)

Actually, as there is (at least) one page that refers to this one for semidirect products of lie algebras (Levi_decomposition), it would be right to define it. I, myself, do not know what it is. I guess that the rules are the ones you get for the lie algebra of a semidirect product of lie groups, but if there is an expert, let him speak.

--YannickSamba (talk) 16:38, 25 January 2012 (UTC)

The Lie algebra and bracket can be found in Lie algebra extension#Semidirect product (groups). YohanN7 (talk) 21:46, 17 May 2015 (UTC)

Inverse

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

Properties

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)

Relations to direct products

This statement

If H is also normal in G, or equivalently, if there exists a homomorphism G → N which is the identity on N

should be replaced by

If H is also normal in G, or equivalently, if there exists a homomorphism G → N which is the identity on N and whose kernel is H.

The kernel is isomorphic to H but this does not mean that H is normal in G.

Confusing introductory paragraph of section Semidirect products and group homomorphisms

The section Semidirect products and group homomorphisms opens with this paragraph:

"Let G be a semidirect product of the normal subgroup N and the subgroup H. Let Aut(N) denote the group of all automorphisms of N. The map φ : H → Aut(N) defined by φ(h) = φ_h, where φ_h(n) = hnh⁻¹ for all h in H and n in N, is a group homomorphism. (Note that hnh⁻¹∈N since N is normal in G.) Together N, H and φ determine G up to isomorphism, as we show now."

But the section immediately continues by defining the semidirect product in terms of an arbitrary homomorphism φ : H → Aut(N). And so the use of the specific homomorphism defined by φ_h(n) = hnh⁻¹ is misleading and confusing. Particularly when it is immediately followed by the sentence "Together N, H and φ determine G up to isomorphism, as we show now."

The homomorphism φ_h(n) = hnh⁻¹ may be an excellent example of such a homomorphism, but that is not how it is described in this paragraph.Daqu (talk) 22:08, 25 July 2014 (UTC)

Actually, if N and H are given/fixed as subgroups inside a group, so is the homomorphism. 86.127.138.67 (talk) 19:07, 7 April 2015 (UTC)

I just hit upon this confusion as well. In the first part, it looks like φ is uniquely defined and the semidirect product is unique. Then later on, it talks about different φ giving different semidirect products with the same N and H. After reading over the page several times, I think I've realised what's going on. Inner semidirect products are unique, but outer semidirect products are not. φ is defined as (h ↦ hnh⁻¹) in inner semidirect products, but in outer semidirect products φ can be any homomorphism from H to Aut(N). Let me explain in more detail:

Inner Semidirect Products

Let G be a group, with subgroup H and normal subgroup N. If NH = G, and N∩H = {e}, then G is the (inner) semidirect product of N and H.

Note that we didn't need to choose a φ. Since we already have the group G, the semidirect product of its subgroups N and H is unique and unambiguous.

Every element of G = N⋊H can be written uniquely as "nh" with n∈N and h∈H.

Thus the product of n₁h₁∈G and n₂h₂∈G is just (n₁h₁)(n₂h₂) = n₁h₁n₂(h₁^-1h₁)h₂ = (n₁h₁n₂h₁^-1)(h₁h₂)

Letting φ_h denote the function (n ↦ hnh⁻¹), i.e. conjugation by h, we can write this as (n₁ φ_h1(n₂))(h₁h₂), which is once again clearly of the form "nh".

Outer Semidirect Products

Let N and H be groups, and let φ : H → Aut(N) be a group homomorphism.

Then we can define the outer semidirect product of N and H with respect to φ, denoted N⋊_φH, as the set of pairs (n,h) with n∈N and h∈H, with the multiplication defined by (n₁,h₁)(n₂,h₂) := (n₁ φ(h₁)(n₂), h₁h₂)

It wouldn't make sense to define φ here as φ(h) := φ_h = (n ↦ hnh⁻¹), because multiplication between elements of N and elements of H hasn't been defined yet! N and H are in different unrelated groups!

However, once we have chosen some φ : H → Aut(N) and defined N⋊_φH, if we then identify n∈N with (n,e_H)∈N⋊_φH and h∈H with (e_N,h)∈N⋊_φH, we can show that:

[ Noting that (n,h)^-1 = (φ(h^-1)(n^-1), h^-1) ]

φ_h(n) = hnh^-1 = (e_N,h)(n,e_H)(e_N,h)^-1 = (e_N,h)(n,e_H)(φ(h^-1)(e_N^-1), h^-1) = (e_N φ(h)(n), he_H)(φ(h^-1)(e_N), h^-1) = (φ(h)(n), h)(e_N, h^-1) = (φ(h)(n) φ(h)(e_N), hh^-1) = (φ(h)(n), e_H) = φ(h)(n)

In summary, in the inner semidirect product, N and H are subgroups of a group G, so multiplication between them is already defined, and so (n₁h₁)(n₂h₂) = (n₁ φ_h1(n₂))(h₁h₂), where φ_h is conjugation by h.
Whereas in the outer semidirect product, N and H are two unrelated groups, so you define (n₁,h₁)(n₂,h₂) := (n₁ φ(h₁)(n₂), h₁h₂) with some arbitrary φ of your choice. Once this is done, multiplication between N and H can be defined by treating N and H as the appropriate subgroups of N⋊_φH, in which case you do find that φ_h(n) = hnh⁻¹ is actually indeed equal to φ(h)(n).

The article does have separate sections for inner semidirect products and outer semidirect products, but the explanation of these two different situations and how they relate to each other is poorly worded, hence the confusion. Hopefully someone can fix this. I might give it a try myself, but I'd rather someone who is more familiar with the topic. --- AndreRD (talk) 21:35, 5 February 2019 (UTC)

Generaliztions in categories

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)