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joostef@gmail.com
science forum beginner
Joined: 13 May 2006
Posts: 5
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 Posted: Tue May 23, 2006 11:35 pm Post subject:
Group theory: Z_n (n approaches infinity)
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Consider Z_n = {0,1,2,...,n-1}, which forms a group under addition (mod
n). So, x+y (mod n) = x+y, if 0<=x+y<n; or x+y-n, if x+y>=n. Now let
n go to infinity and thus define L = Z_n (n -> infinity).
Could anyone help me with the following questions?
(1) Is L a group?
(1b) Since any finite sum x_1+x_2+...+x_k < infinity, that is to say,
any finite sum converges in L, can we say that L is isomorphic to the
non-negative integers? What I mean is, does there exist a bijective
function f : L -> N={0,1,2,...}? If we look at how I defined addition
(mod n), it would appear that we will never have the case x+y >
infinity, so closure will be maintained.
I'm having a bit of trouble wrapping my mind around it. Intuitively,
it makes sense to me, but I know what holds for finite n may not
necessarily hold when we go arbitrarily large. Is there a problem with
my construction?
-Francois. |
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Brian M. Scott
science forum Guru
Joined: 10 May 2005
Posts: 332
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| Posted: Tue May 23, 2006 11:56 pm Post subject:
Re: Group theory: Z_n (n approaches infinity)
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On 23 May 2006 16:35:58 -0700, <joostef@gmail.com> wrote in
<news:1148427358.091912.157420@g10g2000cwb.googlegroups.com>
in alt.math.undergrad:
| Quote: | Consider Z_n = {0,1,2,...,n-1}, which forms a group under addition (mod
n). So, x+y (mod n) = x+y, if 0<=x+y<n; or x+y-n, if x+y>=n. Now let
n go to infinity and thus define L = Z_n (n -> infinity).
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How? This isn't a standard notion of limit, so it isn't
meaningful until you give it a meaning.
[...]
| Quote: | Is there a problem with my construction?
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Yes: it's undefined.
Brian |
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joostef@gmail.com
science forum beginner
Joined: 13 May 2006
Posts: 5
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| Posted: Wed May 24, 2006 1:14 am Post subject:
Re: Group theory: Z_n (n approaches infinity)
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| Quote: | Consider Z_n = {0,1,2,...,n-1}, which forms a group under addition (mod
n). So, x+y (mod n) = x+y, if 0<=x+y<n; or x+y-n, if x+y>=n. Now let
n go to infinity and thus define L = Z_n (n -> infinity).
How? This isn't a standard notion of limit, so it isn't
meaningful until you give it a meaning.
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Hm. Good point. How about this: to say Z_n approaches the limit L as
n approaches infinity means that, for any positive integer K, there is
a positive integer M such that the order of Z_n is greater or equal to
K for all n>=M. Whereas Z_n={m : m nonnegative integer from 0 to n-1},
we would have L={m : m nonnegative integer from 0 to lim(n->infinity)
(n-1)}.
-Francois. |
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Brian M. Scott
science forum Guru
Joined: 10 May 2005
Posts: 332
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| Posted: Wed May 24, 2006 1:30 am Post subject:
Re: Group theory: Z_n (n approaches infinity)
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On 23 May 2006 18:14:46 -0700, <joostef@gmail.com> wrote in
<news:1148433286.668542.38030@i40g2000cwc.googlegroups.com>
in alt.math.undergrad:
| Quote: | Consider Z_n = {0,1,2,...,n-1}, which forms a group under addition (mod
n). So, x+y (mod n) = x+y, if 0<=x+y<n; or x+y-n, if x+y>=n. Now let
n go to infinity and thus define L = Z_n (n -> infinity).
How? This isn't a standard notion of limit, so it isn't
meaningful until you give it a meaning.
Hm. Good point. How about this: to say Z_n approaches the limit L as
n approaches infinity means that, for any positive integer K, there is
a positive integer M such that the order of Z_n is greater or equal to
K for all n>=M.
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I'm afraid that this doesn't make sense as stated: you're
trying to define 'Z_n approaches L as n approaches
infinity', but your attempt at a definition nowhere mentions
L.
You actually need to define a meaningful general notion of
the limit of a sequence of sets before any of this is going
to make sense. There are notions of that kind, but many of
them require a topological context. Here the only
reasonable notion of of the limit of the sets Z_n is their
union, the set N = {0, 1, 2, ...} of natural numbers.
However, the group operations on the Z_n are not related in
any nice way, and as a result there is no natural limit of
these operations on N. Any group operation that you define
on N will be essentially unrelated to the operations on the
Z_n. (Note that ordinary operation absolutely will not
work, since you won't have inverses.)
[...]
Brian |
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joostef@gmail.com
science forum beginner
Joined: 13 May 2006
Posts: 5
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| Posted: Wed May 24, 2006 1:45 am Post subject:
Re: Group theory: Z_n (n approaches infinity)
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Brian wrote:
| Quote: | However, the group operations on the Z_n are not related in
any nice way, and as a result there is no natural limit of
these operations on N. Any group operation that you define
on N will be essentially unrelated to the operations on the
Z_n. (Note that ordinary operation absolutely will not
work, since you won't have inverses.)
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Ah, now I understand. I was doing a lot of handwaving, but I wasn't
even thinking about how the operations relate to each other. And your
last point tells me at least that if L had a similar structure to N,
then, yeah, it wouldn't be a group, which answers my question.
Thanks for the help.
-Francois. |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Wed May 24, 2006 3:50 am Post subject:
Re: Group theory: Z_n (n approaches infinity)
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In article <1148427358.091912.157420@g10g2000cwb.googlegroups.com>,
<joostef@gmail.com> wrote:
| Quote: | Consider Z_n = {0,1,2,...,n-1}, which forms a group under addition (mod
n). So, x+y (mod n) = x+y, if 0<=x+y<n; or x+y-n, if x+y>=n. Now let
n go to infinity and thus define L = Z_n (n -> infinity).
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And what does that mean, exactly?
What concept of limit are you trying to use here? There isn't a unique
one, you know.
In group theory, there ->is<- a concept of limit, but you need to
specify a lot more than just what the groups are.
| Quote: | Could anyone help me with the following questions?
(1) Is L a group?
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Right now, "L" is nonsense.
| Quote: | I'm having a bit of trouble wrapping my mind around it.
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Probably because you are being extremely sloppy.
First, you need to decide just ->what<- it means to take "the limit of
a sequence of groups". What does it mean to say that L is "the limit
of Z_n as n goes to infinity"? To me, it's just nonsense.
| Quote: | Is there a problem with my construction?
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Don't know. You haven't given any constructions. You just threw some
notation at the wall and hoped it would stick. But we're not trying to
cook pasta, you know.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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Arturo Magidin
science forum Guru
Joined: 25 Mar 2005
Posts: 1838
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| Posted: Wed May 24, 2006 3:53 am Post subject:
Re: Group theory: Z_n (n approaches infinity)
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In article <1148433286.668542.38030@i40g2000cwc.googlegroups.com>,
<joostef@gmail.com> wrote:
| Quote: | Consider Z_n = {0,1,2,...,n-1}, which forms a group under addition (mod
n). So, x+y (mod n) = x+y, if 0<=x+y<n; or x+y-n, if x+y>=n. Now let
n go to infinity and thus define L = Z_n (n -> infinity).
How? This isn't a standard notion of limit, so it isn't
meaningful until you give it a meaning.
Hm. Good point. How about this: to say Z_n approaches the limit L as
n approaches infinity means that, for any positive integer K, there is
a positive integer M such that the order of Z_n is greater or equal to
K for all n>=M.
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This still makes no sense. First, note that your "definition"
mentions L at the beginning, but fails to mention it again. So why is
L part of the "definition"?
What you describe is simply: a sequence of finite groups of increasing
order, whose order grows without limit. You have not really defined
any kind of limit, though.
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu |
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