![]() | This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
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)
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)
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)
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:
Of course, in both cases, we write G = N XH. Maybe you misunderstood my statement to mean that N XH is the same as H XN? 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)
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)
If you're only talking about subgroups, I would think this would be the join, , (not the meet, , the intersection is meet) of N and H, see Lattice of subgroups. Summsumm ( talk) 10:20, 20 May 2009 (UTC)
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)
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)
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)
(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)
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)
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 |
AFAIK the idea behind this symbol is that it combines the relations and (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)
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):
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)
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)
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)
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)
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)
I've made a proposal at Wikipedia:WikiProject Mathematics/Conventions. Let's hold further discussion there. -- Fropuff 01:35, 12 January 2006 (UTC)
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)
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
One important example is given by with the operation of matrix multiplication and with the operation of vector addition. Clearly G acts on H in the natural way, i.e. identify 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 with the structure
This reflects the fact that applying a linear transformation will alter any translation that went before. Dharma6662000 ( talk) 19:16, 30 July 2008 (UTC)
Do you think it should be explained differently? Algebraist 23:47, 21 August 2008 (UTC)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 (or simply N ×φ H) is called the semidirect product of N and H with respect to φ, defined as follows.
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)
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)
I dont think the inverse element is right. —Preceding unsigned comment added by 131.174.17.85 ( talk) 14:35, 26 October 2010 (UTC)
The current statement is that the order of 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)
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.
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) = φh, where φh(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 φh(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 φh(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)
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:
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 φh1(n2))(h1h2), where φh 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 φh(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)
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)
![]() | This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
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)
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)
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)
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:
Of course, in both cases, we write G = N XH. Maybe you misunderstood my statement to mean that N XH is the same as H XN? 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)
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)
If you're only talking about subgroups, I would think this would be the join, , (not the meet, , the intersection is meet) of N and H, see Lattice of subgroups. Summsumm ( talk) 10:20, 20 May 2009 (UTC)
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)
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)
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)
(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)
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)
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 |
AFAIK the idea behind this symbol is that it combines the relations and (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)
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):
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)
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)
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)
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)
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)
I've made a proposal at Wikipedia:WikiProject Mathematics/Conventions. Let's hold further discussion there. -- Fropuff 01:35, 12 January 2006 (UTC)
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)
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
One important example is given by with the operation of matrix multiplication and with the operation of vector addition. Clearly G acts on H in the natural way, i.e. identify 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 with the structure
This reflects the fact that applying a linear transformation will alter any translation that went before. Dharma6662000 ( talk) 19:16, 30 July 2008 (UTC)
Do you think it should be explained differently? Algebraist 23:47, 21 August 2008 (UTC)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 (or simply N ×φ H) is called the semidirect product of N and H with respect to φ, defined as follows.
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)
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)
I dont think the inverse element is right. —Preceding unsigned comment added by 131.174.17.85 ( talk) 14:35, 26 October 2010 (UTC)
The current statement is that the order of 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)
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.
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) = φh, where φh(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 φh(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 φh(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)
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:
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 φh1(n2))(h1h2), where φh 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 φh(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)
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)