Talk:Cyclic group

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Contents 1 (no heading) 2 Symmetry groups 3 generator for Z 4 periodic group 5 Uniformization of notation

[edit] (no heading)

Did a nearly universal change in notation take place? Why do most people write nowadays instead of ? Phys 17:02, 30 Aug 2003 (UTC)

Perhaps because Z_p is also used for the p-adic integers. Or perhaps because Z_n requires a special definition, whereas Z/nZ just reuses other standard notation. --Zundark 17:28, 30 Aug 2003 (UTC)

we could mention it's the quotient group notation in the article :) -- Tarquin 20:30, 30 Aug 2003 (UTC)

I've also seen the notation J_n for the finite cyclic groups (where of course J is used for the infinite cyclic group). This avoids the conflict with Z_p as the p-adic integers, but it appears this still leaves some ambiguity as J₁, J₂, J₃ and J₄ are commonly used for the Janko groups. Still, there is an advantage to using J_n or Z_n as it then makes sense to use the same notation for the corresponding finite rings, whereas C_n is unintuitive in that context. CyborgTosser 21:31, 28 Jun 2004 (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 Z_n, 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 'C_n'? This isn't a joke or a dumb question. You say ' for any positive integer n, there is a cyclic group C_n of order n ', but you never actually say precisely what the elements of C_n are. There are many different realisations of the cyclic group with n elements, so it almost seems as if you're using the C_n 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 'C_n is isomorphic Z/nZ', because how are you supposed to have an isomorphism when you haven't even told me what the elements of C_n 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 C_n 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 C_n 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)

That sounds like an acceptable way to do it, since free groups can be proved to be unique up to isomorphism. Is this any different from the group presentation? Revolver 19:50, 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.

Of course, now, (after morning coffee), I realise you must have meant U(1,C), 1-dim unitary group over C, and this is exactly S^1, or SO(2,R), you are correct. Revolver 22:50, 1 Apr 2004 (UTC)

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)

[edit] Symmetry groups

I deleted the following text, added by User:Patrick

The symmetry group for n-fold rotational symmetry is algebraically of type C_n, and itself also denoted by C_n. However, for even order there is another cyclic symmetry group in 3D, which is algebraically the same: the group S_2n generated by a rotation by an angle 180°/n about an axis, combined with a reflection in the plane perpendicular to the axis, is cyclic of order 2n. For order 2 there is a third one: C_s, which contains reflection in a plane; for S₂ the notation S_i is also used; it contains inversion. More generally, for any order 2n which is twice an odd number n, there is a third one; it has an n-fold rotation axis, and a perpendicular plane of reflection: C_nh. It is generated by a rotation by an angle 360°/n about the axis, combined with the reflection.

I didn't understand it at all, and it looked wrong. S_n 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)

I am referring to isometries in 3D space, as explained in Symmetry_group#Three_dimensions, and use the notations used there. Unfortunately S_n is a notation used for two different things, I agree that that is confusing. I will see if there is another suitable notation, otherwise I can at least add a remark about this.

The first two are:

• The symmetry group for n-fold rotational symmetry
• the group S_2n generated by a rotation by an angle 180°/n about an axis, combined with a reflection in the plane perpendicular to the axis

Patrick 17:22, 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)

These are examples, I do not know what is confusing about saying which space we are talking about (the original text was about 2D, but this was only implicit). 4D etc. might be interesting, perhaps somebody can add that.

I moved the 3D part down. Obviously, people who do not know what a symmetry group is can go to that article to learn about that. This section does not try to summarize that whole article, only the parts about the cyclic groups. It is common that if subjects A and B have a connection, this is mentioned in both articles. Only if the common part is rather long, the complete treatment can better be restricted to one of the two articles.--Patrick 07:05, 16 September 2005 (UTC)Patrick 06:48, 16 September 2005 (UTC)

OK, I understand that section now. But, not to be dense, but I don't see what this has to do with "3D". Why not "4D" "11D" or "26D" ? One can define reflections in the plane just fine, without needing recourse to higher dimension. In fact, reflections can be defined just fine without needing recourse to geometry (i.e. without saying "dimension" at all). Finally, since the far more common usage for S_n is the symmetric group, I think its critical to mention that the S_2n here is NOT the symmetric group, and wikilink it to something that defines what it actually is. linas 19:04, 16 September 2005 (UTC)

If you mean that you can describe the cyclic symmetry groups in general for n-dimensional space, go ahead.--Patrick 01:02, 17 September 2005 (UTC)

No, you miss my point. I was trying to say, as politly as possible, that there is no such thing as a "cyclic symmetry group in 3D". They are all the same thing: dimension has nothing to do with the cyclic group. linas 04:13, 17 September 2005 (UTC)

Symmetry groups are groups of isometries, so they depend on the metric, i.e. the geometry.--Patrick 01:35, 17 September 2005 (UTC)

Yes, but the cyclic group, in and of itself, is not a symmetry group. It can be used, if desired, as a symmetry group of n-dimensional space for n>1. However, the cyclic group often occurs in contexts in which there is no dimension whatsoever, e.g. in number theory. Group theory in general, while often related to geometry, is not a sub-branch of geometry.

I know, I do not think I suggested otherwise.--Patrick 12:59, 17 September 2005 (UTC)

This thread of conversation is convincing me that that whole section should be removed in its entirity from this article. I strongly suggest that you start a new article, titled, possibly List of two dimensional symmetry groups, in which you can discuss this topic. linas 04:13, 17 September 2005 (UTC)

Ok, I moved it. However, I find it odd that you get so uncomfortable and confused by an overview of some cyclic groups occurring in geometry.--Patrick 13:15, 17 September 2005 (UTC)

Thank you; it appears to be a marvelous article. My discomfort was two-fold:

• The problem with the S_n notation, which you have fixed.
• The statement that reflection has something to do with 3D, when reflection has the same, identical, definition in any number of dimensions, and can be defined without reference to dimension. This article is non-geometric in nature; the appearance of 3D seemed a poor fit. linas 22:42, 17 September 2005 (UTC)

[edit] generator for Z

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)

It's not a very good argument. A free group on two generators has a comparable property, i.e. the group's automorphisms act transitively on the ordered pairs of generators. But the number of such pairs is very large. (And even for the free abelian group on two generators, there are as many such pairs as invertible 2×2 integer matrices, all 'equivalent' in a sense. That freedom is the modular group!) Charles Matthews 13:10, 24 October 2005 (UTC)

[edit] periodic group

In the article we have

A cyclic group should not be confused with a periodic group.

are not all cyclic groups periodic? --Salix alba (talk) 21:41, 20 March 2006 (UTC)

The finite cyclic groups are periodic (like all finite groups), but the infinite cyclic group isn't periodic. --Zundark

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)

Both terms are standard. I suggest we stick with periodic group, which is what the MSC uses (20F50). --Zundark 09:26, 15 March 2007 (UTC)

[edit] Uniformization of notation

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)

I've never seen Z/n as notation... I'm very familiar with Z/nZ but not what you have written. I'd like to hear from a few more people on waht is standard. - grubber 00:35, 15 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 think if we are going to strike Z_n from the article, I would rather see Z/nZ, since that is a notation everyone will agree on. - grubber 04:25, 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)

Z/nZ means something: take Z and mod out by the normal subgroup nZ. This is notation used in every algebra textbook I've ever seen. Why would you drop the Z from the mod? How do you mod out by an element -- perhaps you mean the subgroup generated by n, but then you would write (n) or <n>. In any case, the notation you are suggesting seems to be far less common that Z/nZ. I don't see why you are saying it should stay. Can you provide the name of an algebra textbook that uses this notation? - grubber 16:47, 15 March 2007 (UTC)

I agree with grubber and I think we should stick to the most common notation. Wikipedia is not the place to decide on the best notation to use.MathMartin 17:30, 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 agree that all notations should be mentioned, and that one should be used throughout. In my own personal work, I prefer Z_n (I don't know much number theory, so I dont get confused :) Given that, I think Z/nZ is the clearest and most unambiguous. - grubber 00:00, 16 March 2007 (UTC)

In fairness, the angle brackets are not wrong-minded. This is an article about groups, and we are modding out by a subgroup, not an ideal. Z may be a commutative ring, but it is also (trivially so) an abelian group. - grubber 02:14, 16 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)

I'm not debating whether it's a valid notation or whether you or anyone else uses it (you gave examples). But, this is the first time I've ever seen Z/n, and I have read Herstein, Lange, Milne, and Ash, as well as some Springer books. I have not read many academic papers in algebra, so I can't testify to what is common there. My point is that as an educational tool Z/nZ is something we both agree is unambiguous, prolific, and notationally precise (it is clear that it's a quotient group and it's clear what the groups are). It seems the most neutral option. - grubber 02:10, 16 March 2007 (UTC)

I agree with G. Kuperburg that uniform notation through the entire article is worthwhile. Z/nZ seems like a good solution since both of you seem willing to accept it. I think that the intended reader here will have little exposure to research papers but will have some exposure to undergraduate textbooks, so sticking to a notation that commonly appears in texts has some advantages. CMummert · talk 18:23, 16 March 2007 (UTC)

From what I have seen: C_n (and also Z_n) are common for the cyclic group; Z/nZ is mainly used when you consider the ring structure and GF_n when it is a field. pom 22:40, 16 March 2007 (UTC)

The discussion in the German article echoes pom's analysis. We tend to see C_n used when the group is multiplicative, as in rotational symmetry. The additive groups of cohomology are often written Z_n, which in that context is clear and concise; however, in number theory the notation Z_p also means the p-adic integers. When we want the ring of integers modulo n, we invariably use the general quotient notation, Z/nZ instead. Of course, quotient groups are as valid as quotient rings. Our Galois field article points out that GF(pⁿ) is not the same as Z/pⁿZ; for example, GF(4) is (group) isomorphic to Z₂×Z₂, which is not Z₄.

In kindness to our readers, the preceeding facts should be on the article page, not just the talk page.

Amongst ourselves, we know that notation is a protean tool, constantly adapting to the needs of the moment. Clearly the cyclic groups (and related rings and fields) appear naturally in many contexts; so if an algebraic geometer, say, is accustomed to writing a principal ideal as (a), then we must be prepared to accept the associated quotient. And if some experts find it expedient to trim the notation down to Z/n, we can hardly be shocked. But readers sophisticated enough to read those papers can be expected to cope; we need not burden our average Wikipedia inquirer with this notation until it becomes widespread.

A typographical nicety: In contexts where angle brackets are deemed appropriate, the correct markup uses "&lang;" and "&rang;", as in 〈n〉, not "&lt;" ("<") and "&gt;" (">"), as in <n>.

Finally, my personal preference is for Z_n (or C_n, in context), which for me is concise, clear, and familiar; but I have learned to live with the more cumbersome Z/nZ when I must. The abbreviated Z/n seems like a creative compromise; I'll be happy to use it here if it catches on more widely. --KSmrq^T 04:35, 17 March 2007 (UTC)

C_n seems reasonable to me as well. - grubber 23:19, 17 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 Z_n is increasingly problematic, because in fact p-adic coefficients are sometimes important in cohomology, for example in etale cohomology. In addition Z_n (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)

My point is that Z/n is not as common or standard as the others. The English WP article on cyclic groups is the only one that even mentions it as an alternative notation, and that's because you added it. The foreign language WP articles are about evenly mixed between Z_n and Z/nZ (many also using C_n tho). This is a pretty poor survey population to make much of a popularity conclusion on, but it's still worth mentioning. - grubber 23:30, 17 March 2007 (UTC)