basic doubt about groups

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alessandro95
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basic doubt about groups

Postby alessandro95 » Thu Aug 15, 2013 5:00 pm UTC

I'm reading, out of personal interest, an introductory book to abstract algebra and I've just finished the chapter about cyclic groups. The book says every cyclic group is also abelian and also provides a simple proof of this fact but it doesn't say anything about the opposite so I was wondering if every abelian group is also cyclic; that intuitively lools wrong.to me but I cannot find any example of an.abelian group that isn't cyclic, can somebody provide one?
The primary reason Bourbaki stopped writing books was the realization that Lang was one single person.

Nitrodon
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Re: basic doubt about groups

Postby Nitrodon » Thu Aug 15, 2013 5:42 pm UTC

The smallest non-cyclic group is the Klein four-group, which happens to be abelian.

lalop
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Re: basic doubt about groups

Postby lalop » Thu Aug 15, 2013 9:37 pm UTC

Linking to wiki may not be the most illustrative in this case, IMO. Consider the direct product Z/2 x Z/2, which is abelian as the direct product of abelian groups. Then how would you generate the element (1,1)? The only other elements, (0,0), (1,0), (0,1) clearly do not generate it, so they can't be cyclic generators, and (1,1) does not generate the whole group, so it can't be a cyclic generator either. Therefore, Z/2 x Z/2 is not cyclic.

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MartianInvader
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Re: basic doubt about groups

Postby MartianInvader » Thu Aug 15, 2013 10:07 pm UTC

If you're very new to group theory, some more-familiar examples might be the rational, real, or complex numbers under addition, or the nonzero rational, real, or complex numbers under multiplication.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

jedelmania
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Re: basic doubt about groups

Postby jedelmania » Thu Aug 15, 2013 11:37 pm UTC

MartianInvader wrote:If you're very new to group theory, some more-familiar examples might be the rational, real, or complex numbers under addition, or the nonzero rational, real, or complex numbers under multiplication.


This is also what I'd recommend. Other, slightly more exotic examples include polynomials under addition, subsets of a set under symmetric difference and {a+b sqrt(2)} under addition.

If you are starting to learn about groups, one important thing to remember is that groups crop up in so many places, in addition, multiplication, many set operations and geometric transformations (such as rotation, enlargement and translation). If you prove a property in group theory, you are proving a property about a huge number of important mathematical operations.

tomtom2357
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Re: basic doubt about groups

Postby tomtom2357 » Thu Aug 22, 2013 5:06 am UTC

However, every finite abelian group is a product of cyclic groups.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

RedJelloNinja
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Re: basic doubt about groups

Postby RedJelloNinja » Thu Aug 29, 2013 2:25 am UTC

I'm sure you meant every finitely generated abelian group is the product of cyclic groups.

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Re: basic doubt about groups

Postby tomtom2357 » Thu Aug 29, 2013 3:08 am UTC

I guess your statement is slightly stronger than mine. However, it is easier for someone (in my opinion) who has just started group theory to understand "finite" than "finitely generated"
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

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jestingrabbit
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Re: basic doubt about groups

Postby jestingrabbit » Thu Aug 29, 2013 3:54 am UTC

RedJelloNinja wrote:I'm sure you meant every finitely generated abelian group is the product of cyclic groups.


The integers under addition are finitely generated and are not a product of cyclic groups.
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tomtom2357
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Re: basic doubt about groups

Postby tomtom2357 » Thu Aug 29, 2013 4:08 am UTC

I think that, technically, that would be the infinite cyclic group.
Wikipedia wrote:Any infinite cyclic group is isomorphic to Z, the integers with addition as the group operation.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

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jestingrabbit
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Re: basic doubt about groups

Postby jestingrabbit » Thu Aug 29, 2013 5:03 am UTC

Assuming that everyone is on board with that definition is fraught with misunderstandings.
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Re: basic doubt about groups

Postby tomtom2357 » Thu Aug 29, 2013 5:17 am UTC

It makes the theorems easier. For example, I think it is easier to say: All finitely generated abelian groups are a product of cyclic groups, than: All finitely generated groups are a product of cyclic groups, and some power of the group of integers under multiplication. It is just like how we exclude 1 from being a prime number: To make the theorems easier to state. Anyway, it fits into the definition of a cyclic group:
Wikipedia wrote:In algebra, a cyclic group is a group that is generated by a single element, in the sense that every element of the group can be written as a power of some particular element g in multiplicative notation, or as a multiple of g in additive notation.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

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MartianInvader
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Re: basic doubt about groups

Postby MartianInvader » Thu Aug 29, 2013 9:35 pm UTC

Wait, there are schools of thought out there in which the integers aren't a cyclic group?
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

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jestingrabbit
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Re: basic doubt about groups

Postby jestingrabbit » Thu Aug 29, 2013 10:58 pm UTC

MartianInvader wrote:Wait, there are schools of thought out there in which the integers aren't a cyclic group?


You've got to admit its the freak cyclic group if its a cyclic group. Here's a theorem which reads easier without the freak.

"The number of distinct generators that a cyclic group G is equal to phi(|G|) where phi is the totient function."

Nice for the non freak cyclic groups, meaningless for the freak.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

tomtom2357
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Re: basic doubt about groups

Postby tomtom2357 » Fri Aug 30, 2013 1:13 am UTC

Well, maybe my "all finite abelian groups are a product of cyclic groups" is better then. I don't think anyone would have a reasonable objection to that.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

alessandro95
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Re: basic doubt about groups

Postby alessandro95 » Fri Aug 30, 2013 7:02 am UTC

I understood next to nothing of your last post, and I'm having trouble understanding what exactly is a finitely generated group, wiki wasn't that helpful either, can someone explain this to me?
Thanks!
The primary reason Bourbaki stopped writing books was the realization that Lang was one single person.

tomtom2357
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Re: basic doubt about groups

Postby tomtom2357 » Fri Aug 30, 2013 7:10 am UTC

Every element in a group is expressible in terms of the generators. For example, in the group Z22 (the klein-4-group), the elements (1,0) and (0,1) generate the group because every element can be expressed in terms of those two elements (for example (1,1)=(1,0)+(0,1)). A finitely generated group is a group that can be expressed using finitely many generators.
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Cauchy
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Re: basic doubt about groups

Postby Cauchy » Wed Sep 04, 2013 7:29 pm UTC

jestingrabbit wrote:The integers under addition are finitely generated and are not a product of cyclic groups.


I'm genuinely curious what definition of cyclic you use that excludes Z, because it can't be "a group other than Z which is generated by one element".
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.

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jestingrabbit
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Re: basic doubt about groups

Postby jestingrabbit » Wed Sep 04, 2013 7:51 pm UTC

Cauchy wrote:
jestingrabbit wrote:The integers under addition are finitely generated and are not a product of cyclic groups.


I'm genuinely curious what definition of cyclic you use that excludes Z, because it can't be "a group other than Z which is generated by one element".


You can bake the finiteness into the definition ie define the set \{ C_n :\ n\in\N\} where C_n is defined as you'd expect, then talk about groups being cyclic if they are isomorphic to some C_n.

You've gotta agree that the theory is quite different for the two cases, no?
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

DavCrav
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Re: basic doubt about groups

Postby DavCrav » Fri Sep 06, 2013 9:25 pm UTC

Cauchy wrote:
jestingrabbit wrote:The integers under addition are finitely generated and are not a product of cyclic groups.


I'm genuinely curious what definition of cyclic you use that excludes Z, because it can't be "a group other than Z which is generated by one element".


E.g., a group G is cyclic if there exists x in the group such that G={x,x^2,x^3,...}.


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