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Did a nearly universal change in notation take place? Why do most people write nowadays instead of ? Phys 17:02, 30 Aug 2003 (UTC)
The remark about the discrete logarithm surely belongs in the final section, not with Z/nZ as additive group.
Charles Matthews 12:11, 18 Nov 2003 (UTC)
I always prefer Z/nZ for the cyclic group, rather than Zn, because I work in number theory, and when n is prime, the latter means the ring of p-adic integers, not the cyclic group with p elements. The former notation is never ambiguous. Revolver
What is your definition of 'Cn'? This isn't a joke or a dumb question. You say ' for any positive integer n, there is a cyclic group Cn of order n ', but you never actually say precisely what the elements of Cn are. There are many different realisations of the cyclic group with n elements, so it almost seems as if you're using the Cn notation to denote the entire equivalence class of all groups isomorphic to a cyclic group with n elements. I know this probably sounds ridiculous, but there comes a problem when you say 'Cn is isomorphic Z/nZ', because how are you supposed to have an isomorphism when you haven't even told me what the elements of Cn are? This is something a lot of algebra textbooks do with respect to the cyclic groups, and it drives me nuts. In case you're wondering what possible solutions I have in mind, you could define Cn in terms of generators and relations (in essence, a presentation) or you could define it as the set of integers {0, 1, ..., n − 1} with operation given by remainder of the sum after dividing by n. Either one is independent of the construction as a quotient group of Z, so then it makes sense to say they're isomorphic. Revolver 21:37, 28 Jan 2004 (UTC)
Okay, I see you say that C_n is 'represented by the symmetries of the regular n-gon'. This is almost what I was asking for, except if this is how you want to define C_n, you should just SAY that C_n IS the (rigid motion!) symmetries of the regular n-gon, not that C_n is 'represented' by the motions.
In any case, I don't much care for the C_n notation, mainly for the confusion that we have here...the realisation as the quotient group of the integers is clearly a fixed representative in the isomorphism class, while 'C_n' gives no clue as to what C_n actually is. Revolver 21:45, 28 Jan 2004 (UTC)
Hi, me again...sorry to be a nuisance. I was looking around, I notice that a lot of articles use the C_n notation, esp. in the context of rigid motions symmetry groups and group presentations. So, it seems (imo) the best way to go is to either define C_n as the symmetries of the n-gon, then note that this is isomorphic to presentation ({x},{x^n}), or else to define it as the presentation and note this is isomorphic to symmetries. The same thing could be done for the dihedral groups. Which solution you like probably depends on whether you think of these groups as given by generators and relations, or as symmetries of geometries figures. But at least one choice needs to be made. Once a choice is made, you can show they're isomorphic and agree to use the same notation for each. But you can't slip in C_n in the back door without defining it, say that it's isomorphic to one of the 2 choices, then show they're both isomorphic. You need to start off by explicitly defining it as one or the other. Revolver 21:56, 28 Jan 2004 (UTC)
Isn't it best to define Cn as the quotient of 'the' free group on one generator, by its subgroup of index n? This impacts on the infinite cyclic group, so start with that?
Charles Matthews 17:02, 30 Jan 2004 (UTC)
Not in any fundamental way.
Charles Matthews 20:25, 30 Jan 2004 (UTC)
In Examples of cyclic groups: The group of rotations in a circle, S1, is not a cyclic group.
My 2 questions are: Isnt this group named U(1)? And if it is not cyclic, so what is it then?
200.154.215.124 01:58, 1 Apr 2004 (UTC)
U(1,R) would be the group of all 1x1 orthogonal real matrices, i.e. the group {I, -I} consisting of just the identity matrix and its inverse.
Geometrically, this is the isometries of the line preserving the origin -- there are only two of them, -I representing the "flip" across the origin. I think the group you may be thinking of is SO(2,R), the special orthogonal group of all rotations of the plane fixing the origin. This group is isomorphic to S^1, the group mentioned in the article here, group of rotations of the circle, which is isomorphic to the multiplicative group of all complex numbers of absolute value one. (the unit circle) Multiplication of complex numbers corresponds to rotation in the unit circle, corresponds to addition of angle measure in the matrix representation (cos θ sin θ | −sin θ cos θ), and here is where we can get a fundamental description of S^1 -- since the trig functions have period 2π, S^1 is isomorphic to R under addition, modulo 2π, since the value of the modulus doesn't matter, we can finally say that S^1 is isomorphic to R/Z under addition. This group is uncountable, so it can't possibly be cyclic. Revolver 02:19, 1 Apr 2004 (UTC)
I deleted the following text, added by User:Patrick
I didn't understand it at all, and it looked wrong. Sn is the permutation group, and its completely different. No clue what "3D'" has to do with anything. And if something is a "third one" what were the first two? linas 00:18, 15 September 2005 (UTC)
Ahh, well, a suggestion then: craft a sentance along the lines of "cyclic groups also appear in the theory of crystallographic groups, see the article symmetry group for additional details." The reason for this is two fold: the "standard cyclic group" that is treated here has little to do with 2D or 3D, so its confusing to suddenly talk about dimensions. What about 4D? 5D? Other topolgies? ?? The other problem was that you were trying to summarize in a half dozen sentances a much longer article; instead of doing that, just reference the longer article. linas 04:57, 16 September 2005 (UTC)
Thank you; it appears to be a marvelous article. My discomfort was two-fold:
I believe that 1 generates Z entirely and, consequently, -1 is needless as a generator, since 1^-1 gives -1. Of course, -1 can generate Z as well, but this fact is taken into account by saying that -1 can be mapped to 1. So, so to speak, there exists a unique generator for infinite cyclic groups up to isomorphism. -- Taku 12:09, 24 October 2005 (UTC)
In the article we have
are not all cyclic groups periodic? -- Salix alba ( talk) 21:41, 20 March 2006 (UTC)
I think that "torsion group" is a better and more standard name than "periodic group". The other page should be changed. Greg Kuperberg 21:39, 14 March 2007 (UTC)
I want to give people a heads up on a change that I just made. The notation Z/n is also reasonably widely used nowadays. In my view it is the best choice for the reasons that I wrote on the page itself. In any case the page should have consistent notation, so I went through from beginning to end to uniformize usage. Greg Kuperberg 21:30, 14 March 2007 (UTC)
Z/n is not as common as Z/nZ, but it certainly is a standard notation. See for example here or here. I do not think that Wikipedia needs to strictly adhere to the most common notation. It is enough to mention all common notations and make good use of the best one. Greg Kuperberg 02:37, 15 March 2007 (UTC)
I don't see any real rationale for objecting to Z/n. It is widely used, and it's perfectly clear to any mathematician, even if other notations may be more common. I argue that it's the clearest choice. Greg Kuperberg 04:42, 15 March 2007 (UTC)
I do not know about textbooks, but I have given you examples of real research papers that use this notation. I do indeed mean modding out by the subgroup or ideal generated by the element. You don't strictly need the parentheses because there is nothing else that it could reasonably mean. (Angle brackets are wrong-minded because Z is a commutative ring; describing the subgroup as an ideal is really better.)
I think that Wikipedia is the place to decide the notation that is the clearest for its readers. Certainly all commonly used notations should be mentioned. In my view it is perfectly reasonable to rely on any standard notation that works the best for the audience. I like Z/n for three reasons: It has the brevity of Z_n; it reads the same way that the rings is described verbally; and it has no conflict with p-adic numbers, which I consider a serious concern. But I understand that there is a balance between clarity and orthodoxy. If you feel that what I put is too radical, then it could be reasonable to change the notation to Z/nZ, as long as you do change it consistently from beginning to end --- the article had inconsistent notation before --- and as long as you duly acknowledge all justified notations. Greg Kuperberg 22:28, 15 March 2007 (UTC)
I do not mean to pull rank, but since we have gotten onto personal practices and impressions, I am a mathematics professor with about 40 research papers, and I like to write Z/n. I can also find other mathematicians who I have never met who do the same. You can trust me that I am describing a mainstream viewpoint, although I grant that it's not the only viewpoint. Greg Kuperberg 00:54, 16 March 2007 (UTC)
Certainly one reason that authors might use Z/n in research papers is that it does work as a creative compromise. As for cohomology coefficients, the notation Zn is increasingly problematic, because in fact p-adic coefficients are sometimes important in cohomology, for example in etale cohomology. In addition Zn (if it is not bold or blackboard bold) blurs the distinction between additive and multiplicative group laws. This is sometimes okay, but in an algebra context I find it annoying. Greg Kuperberg 08:32, 17 March 2007 (UTC)
In the definition section I would be tempted to add after G is defined the notation that , otherwise despite G being a group we do not know anything about how it behaves (with repect to closure under multiplication)? 138.38.32.31 ( talk) 18:53, 13 February 2008 (UTC)
What part needs to be translated from the german? The translation request is fairly old, Jan 2005. Should it be removed? JackSchmidt ( talk) 19:37, 13 February 2008 (UTC)
After looking through Google, I can't find much evidence that the terminology “monogenous group” is a serious alternative to “cyclic group”. Some points:
In addition, I am a professional group theorist and I have never heard the “monogenous” terminology before, either in talks or in any article or book that I can recall. I am therefore removing “or monogenous group” from the beginning of this article. Jim ( talk) 17:38, 12 April 2009 (UTC)
Is it true that "For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic"? Is it sufficient that G is a group for g6 = g0? Is G = {1, 2, 4, 8, 16, 32} not a group? 211.30.171.128 ( talk) 05:33, 15 May 2011 (UTC)
It currently says "For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic. In fact, G is essentially the same as (that is, isomorphic to) the set { 0, 1, 2, 3, 4, 5 } with addition modulo 6." That second sentence does not seem to be necessarily correct. Nowhere it says, for example, that g^3 != g^0. If g is taken to be the integer 2 mod 3, then G as stated above is a cyclic group, but it's not isomorphic to Z/6Z. There still is a homomorphism from Z/6Z to G of course. — Preceding unsigned comment added by 192.150.186.171 ( talk) 21:04, 7 September 2011 (UTC)
He appears to be a published mathematician, but in category theory, rather than in group theory. His group theory course notes would only be a valid reference if he's published in group theory. (And I don't see the benefit of the added section, even if it were adequately sourced.) — Arthur Rubin (talk) 19:03, 4 August 2012 (UTC)
Correct me if I'm wrong, but is the following statement necessarily true?
"For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic."
What grounds are that based on? if G = D3, then G is not abelian. Hence, it is not cyclic. Just curious if I'm missing something here before I change it. Pbroks13 ( talk) 20:21, 18 September 2012 (UTC)
I'm having trouble with this claim from the page: "If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler totient function."
There are phi(n) invertible elements, but there are not phi(n) generators. For example in (Z_5, *) there are 4 invertible elements, but only 2 and 3 are generators. — Preceding unsigned comment added by Tbh289 ( talk • contribs) 16:35, 29 December 2012 (UTC)
This point has been alluded to in a thread above. The article states near the end of Cyclic group#Definition:
It would seem to me that it would be natural to use the notation C0 for the infinite cyclic group, since then many statements become more regular, e.g. that Cn is isomorphic to the additive group of Z/nZ (including specifically that that C0 is isomorphic to the additive group of Z/0Z). To use the notation C∞ (or Z∞) strikes me as discarding a lot of the richness of the notation Cn (or equivalent) for the cyclic groups, only to support the interpretation that the subscript denotes the order n of the group (a precedent: the characteristic of a ring). (See [1] for an example of Z0.) The notation of an unsubscripted C or Z may be dominant, but here we are giving the subscripted alternative. The question of course is: which subscripted notation (C∞ or C0) is supported in the literature, so that we can correctly reflect it in the sentence of article quoted here? — Quondum 11:55, 30 September 2013 (UTC)
For some reason this article was already listed as B class even though it largely consisted of a disorganized and repetitious list of facts and there were almost no inline sources. Anyway, I have mostly completed a reorganization of the article, which I hope really does bring it up to something closer to B class. My changes included addition of many inline sources (mostly textbooks), addition of some material (especially Cayley graphs) and removal of some other material (e.g. the cycle graphs of groups, something that appears only in a few sources and not in the standard texts). The one part I haven't done much about is the paragraph about representation theory, now a subsection by itself. It's not an aspect of group theory that I understand very well myself, and currently it has no sources. If someone else wants to take care of properly sourcing that part (and/or checking that what we say there makes sense and hits the important points) I'd be grateful. More eyes everywhere else in the article (and tagging of facts elsewhere that are still not properly sourced) would also be welcome, of course. — David Eppstein ( talk) 00:18, 27 November 2013 (UTC)
I might be wrong, but it seems to me that the generators in the Paley Graph example (figure about the end of the article) has generators 1, 3 and 4. It's written "1,4 and 5".
Cédric VAN ROMPAY ( talk) 17:56, 9 April 2014 (UTC)
This page is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Did a nearly universal change in notation take place? Why do most people write nowadays instead of ? Phys 17:02, 30 Aug 2003 (UTC)
The remark about the discrete logarithm surely belongs in the final section, not with Z/nZ as additive group.
Charles Matthews 12:11, 18 Nov 2003 (UTC)
I always prefer Z/nZ for the cyclic group, rather than Zn, because I work in number theory, and when n is prime, the latter means the ring of p-adic integers, not the cyclic group with p elements. The former notation is never ambiguous. Revolver
What is your definition of 'Cn'? This isn't a joke or a dumb question. You say ' for any positive integer n, there is a cyclic group Cn of order n ', but you never actually say precisely what the elements of Cn are. There are many different realisations of the cyclic group with n elements, so it almost seems as if you're using the Cn notation to denote the entire equivalence class of all groups isomorphic to a cyclic group with n elements. I know this probably sounds ridiculous, but there comes a problem when you say 'Cn is isomorphic Z/nZ', because how are you supposed to have an isomorphism when you haven't even told me what the elements of Cn are? This is something a lot of algebra textbooks do with respect to the cyclic groups, and it drives me nuts. In case you're wondering what possible solutions I have in mind, you could define Cn in terms of generators and relations (in essence, a presentation) or you could define it as the set of integers {0, 1, ..., n − 1} with operation given by remainder of the sum after dividing by n. Either one is independent of the construction as a quotient group of Z, so then it makes sense to say they're isomorphic. Revolver 21:37, 28 Jan 2004 (UTC)
Okay, I see you say that C_n is 'represented by the symmetries of the regular n-gon'. This is almost what I was asking for, except if this is how you want to define C_n, you should just SAY that C_n IS the (rigid motion!) symmetries of the regular n-gon, not that C_n is 'represented' by the motions.
In any case, I don't much care for the C_n notation, mainly for the confusion that we have here...the realisation as the quotient group of the integers is clearly a fixed representative in the isomorphism class, while 'C_n' gives no clue as to what C_n actually is. Revolver 21:45, 28 Jan 2004 (UTC)
Hi, me again...sorry to be a nuisance. I was looking around, I notice that a lot of articles use the C_n notation, esp. in the context of rigid motions symmetry groups and group presentations. So, it seems (imo) the best way to go is to either define C_n as the symmetries of the n-gon, then note that this is isomorphic to presentation ({x},{x^n}), or else to define it as the presentation and note this is isomorphic to symmetries. The same thing could be done for the dihedral groups. Which solution you like probably depends on whether you think of these groups as given by generators and relations, or as symmetries of geometries figures. But at least one choice needs to be made. Once a choice is made, you can show they're isomorphic and agree to use the same notation for each. But you can't slip in C_n in the back door without defining it, say that it's isomorphic to one of the 2 choices, then show they're both isomorphic. You need to start off by explicitly defining it as one or the other. Revolver 21:56, 28 Jan 2004 (UTC)
Isn't it best to define Cn as the quotient of 'the' free group on one generator, by its subgroup of index n? This impacts on the infinite cyclic group, so start with that?
Charles Matthews 17:02, 30 Jan 2004 (UTC)
Not in any fundamental way.
Charles Matthews 20:25, 30 Jan 2004 (UTC)
In Examples of cyclic groups: The group of rotations in a circle, S1, is not a cyclic group.
My 2 questions are: Isnt this group named U(1)? And if it is not cyclic, so what is it then?
200.154.215.124 01:58, 1 Apr 2004 (UTC)
U(1,R) would be the group of all 1x1 orthogonal real matrices, i.e. the group {I, -I} consisting of just the identity matrix and its inverse.
Geometrically, this is the isometries of the line preserving the origin -- there are only two of them, -I representing the "flip" across the origin. I think the group you may be thinking of is SO(2,R), the special orthogonal group of all rotations of the plane fixing the origin. This group is isomorphic to S^1, the group mentioned in the article here, group of rotations of the circle, which is isomorphic to the multiplicative group of all complex numbers of absolute value one. (the unit circle) Multiplication of complex numbers corresponds to rotation in the unit circle, corresponds to addition of angle measure in the matrix representation (cos θ sin θ | −sin θ cos θ), and here is where we can get a fundamental description of S^1 -- since the trig functions have period 2π, S^1 is isomorphic to R under addition, modulo 2π, since the value of the modulus doesn't matter, we can finally say that S^1 is isomorphic to R/Z under addition. This group is uncountable, so it can't possibly be cyclic. Revolver 02:19, 1 Apr 2004 (UTC)
I deleted the following text, added by User:Patrick
I didn't understand it at all, and it looked wrong. Sn is the permutation group, and its completely different. No clue what "3D'" has to do with anything. And if something is a "third one" what were the first two? linas 00:18, 15 September 2005 (UTC)
Ahh, well, a suggestion then: craft a sentance along the lines of "cyclic groups also appear in the theory of crystallographic groups, see the article symmetry group for additional details." The reason for this is two fold: the "standard cyclic group" that is treated here has little to do with 2D or 3D, so its confusing to suddenly talk about dimensions. What about 4D? 5D? Other topolgies? ?? The other problem was that you were trying to summarize in a half dozen sentances a much longer article; instead of doing that, just reference the longer article. linas 04:57, 16 September 2005 (UTC)
Thank you; it appears to be a marvelous article. My discomfort was two-fold:
I believe that 1 generates Z entirely and, consequently, -1 is needless as a generator, since 1^-1 gives -1. Of course, -1 can generate Z as well, but this fact is taken into account by saying that -1 can be mapped to 1. So, so to speak, there exists a unique generator for infinite cyclic groups up to isomorphism. -- Taku 12:09, 24 October 2005 (UTC)
In the article we have
are not all cyclic groups periodic? -- Salix alba ( talk) 21:41, 20 March 2006 (UTC)
I think that "torsion group" is a better and more standard name than "periodic group". The other page should be changed. Greg Kuperberg 21:39, 14 March 2007 (UTC)
I want to give people a heads up on a change that I just made. The notation Z/n is also reasonably widely used nowadays. In my view it is the best choice for the reasons that I wrote on the page itself. In any case the page should have consistent notation, so I went through from beginning to end to uniformize usage. Greg Kuperberg 21:30, 14 March 2007 (UTC)
Z/n is not as common as Z/nZ, but it certainly is a standard notation. See for example here or here. I do not think that Wikipedia needs to strictly adhere to the most common notation. It is enough to mention all common notations and make good use of the best one. Greg Kuperberg 02:37, 15 March 2007 (UTC)
I don't see any real rationale for objecting to Z/n. It is widely used, and it's perfectly clear to any mathematician, even if other notations may be more common. I argue that it's the clearest choice. Greg Kuperberg 04:42, 15 March 2007 (UTC)
I do not know about textbooks, but I have given you examples of real research papers that use this notation. I do indeed mean modding out by the subgroup or ideal generated by the element. You don't strictly need the parentheses because there is nothing else that it could reasonably mean. (Angle brackets are wrong-minded because Z is a commutative ring; describing the subgroup as an ideal is really better.)
I think that Wikipedia is the place to decide the notation that is the clearest for its readers. Certainly all commonly used notations should be mentioned. In my view it is perfectly reasonable to rely on any standard notation that works the best for the audience. I like Z/n for three reasons: It has the brevity of Z_n; it reads the same way that the rings is described verbally; and it has no conflict with p-adic numbers, which I consider a serious concern. But I understand that there is a balance between clarity and orthodoxy. If you feel that what I put is too radical, then it could be reasonable to change the notation to Z/nZ, as long as you do change it consistently from beginning to end --- the article had inconsistent notation before --- and as long as you duly acknowledge all justified notations. Greg Kuperberg 22:28, 15 March 2007 (UTC)
I do not mean to pull rank, but since we have gotten onto personal practices and impressions, I am a mathematics professor with about 40 research papers, and I like to write Z/n. I can also find other mathematicians who I have never met who do the same. You can trust me that I am describing a mainstream viewpoint, although I grant that it's not the only viewpoint. Greg Kuperberg 00:54, 16 March 2007 (UTC)
Certainly one reason that authors might use Z/n in research papers is that it does work as a creative compromise. As for cohomology coefficients, the notation Zn is increasingly problematic, because in fact p-adic coefficients are sometimes important in cohomology, for example in etale cohomology. In addition Zn (if it is not bold or blackboard bold) blurs the distinction between additive and multiplicative group laws. This is sometimes okay, but in an algebra context I find it annoying. Greg Kuperberg 08:32, 17 March 2007 (UTC)
In the definition section I would be tempted to add after G is defined the notation that , otherwise despite G being a group we do not know anything about how it behaves (with repect to closure under multiplication)? 138.38.32.31 ( talk) 18:53, 13 February 2008 (UTC)
What part needs to be translated from the german? The translation request is fairly old, Jan 2005. Should it be removed? JackSchmidt ( talk) 19:37, 13 February 2008 (UTC)
After looking through Google, I can't find much evidence that the terminology “monogenous group” is a serious alternative to “cyclic group”. Some points:
In addition, I am a professional group theorist and I have never heard the “monogenous” terminology before, either in talks or in any article or book that I can recall. I am therefore removing “or monogenous group” from the beginning of this article. Jim ( talk) 17:38, 12 April 2009 (UTC)
Is it true that "For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic"? Is it sufficient that G is a group for g6 = g0? Is G = {1, 2, 4, 8, 16, 32} not a group? 211.30.171.128 ( talk) 05:33, 15 May 2011 (UTC)
It currently says "For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic. In fact, G is essentially the same as (that is, isomorphic to) the set { 0, 1, 2, 3, 4, 5 } with addition modulo 6." That second sentence does not seem to be necessarily correct. Nowhere it says, for example, that g^3 != g^0. If g is taken to be the integer 2 mod 3, then G as stated above is a cyclic group, but it's not isomorphic to Z/6Z. There still is a homomorphism from Z/6Z to G of course. — Preceding unsigned comment added by 192.150.186.171 ( talk) 21:04, 7 September 2011 (UTC)
He appears to be a published mathematician, but in category theory, rather than in group theory. His group theory course notes would only be a valid reference if he's published in group theory. (And I don't see the benefit of the added section, even if it were adequately sourced.) — Arthur Rubin (talk) 19:03, 4 August 2012 (UTC)
Correct me if I'm wrong, but is the following statement necessarily true?
"For example, if G = { g0, g1, g2, g3, g4, g5 } is a group, then g6 = g0, and G is cyclic."
What grounds are that based on? if G = D3, then G is not abelian. Hence, it is not cyclic. Just curious if I'm missing something here before I change it. Pbroks13 ( talk) 20:21, 18 September 2012 (UTC)
I'm having trouble with this claim from the page: "If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler totient function."
There are phi(n) invertible elements, but there are not phi(n) generators. For example in (Z_5, *) there are 4 invertible elements, but only 2 and 3 are generators. — Preceding unsigned comment added by Tbh289 ( talk • contribs) 16:35, 29 December 2012 (UTC)
This point has been alluded to in a thread above. The article states near the end of Cyclic group#Definition:
It would seem to me that it would be natural to use the notation C0 for the infinite cyclic group, since then many statements become more regular, e.g. that Cn is isomorphic to the additive group of Z/nZ (including specifically that that C0 is isomorphic to the additive group of Z/0Z). To use the notation C∞ (or Z∞) strikes me as discarding a lot of the richness of the notation Cn (or equivalent) for the cyclic groups, only to support the interpretation that the subscript denotes the order n of the group (a precedent: the characteristic of a ring). (See [1] for an example of Z0.) The notation of an unsubscripted C or Z may be dominant, but here we are giving the subscripted alternative. The question of course is: which subscripted notation (C∞ or C0) is supported in the literature, so that we can correctly reflect it in the sentence of article quoted here? — Quondum 11:55, 30 September 2013 (UTC)
For some reason this article was already listed as B class even though it largely consisted of a disorganized and repetitious list of facts and there were almost no inline sources. Anyway, I have mostly completed a reorganization of the article, which I hope really does bring it up to something closer to B class. My changes included addition of many inline sources (mostly textbooks), addition of some material (especially Cayley graphs) and removal of some other material (e.g. the cycle graphs of groups, something that appears only in a few sources and not in the standard texts). The one part I haven't done much about is the paragraph about representation theory, now a subsection by itself. It's not an aspect of group theory that I understand very well myself, and currently it has no sources. If someone else wants to take care of properly sourcing that part (and/or checking that what we say there makes sense and hits the important points) I'd be grateful. More eyes everywhere else in the article (and tagging of facts elsewhere that are still not properly sourced) would also be welcome, of course. — David Eppstein ( talk) 00:18, 27 November 2013 (UTC)
I might be wrong, but it seems to me that the generators in the Paley Graph example (figure about the end of the article) has generators 1, 3 and 4. It's written "1,4 and 5".
Cédric VAN ROMPAY ( talk) 17:56, 9 April 2014 (UTC)