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 Post subject: cyclic group
PostPosted: Sep 29, 2007 - 10:26 AM 
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~ Tuổi Thơ ~
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Given the fact that U(49) is cyclic and has 42 elements, deduce the number of generators that U (49) has without actually finding any of the generators.
What do they want?:spider:
List out the elements that are not the generators?

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PostPosted: Sep 29, 2007 - 01:47 PM 
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U(49) là group gì vậy si muội :?

basically, they are asking you how many generators (without finding the generators)

here is a proposition

Let G be a finite cyclic group of order n. Then G = iff gcd(a,n) = 1.
in particular, the number of generators of G is phi(n); phi is the Euler's function

using this proposition, the number of generators of U(49) is phi(42)

see if you can prove the proposition

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 Post subject:
PostPosted: Sep 29, 2007 - 05:50 PM 
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U(49) under multiplication modulo 49 ...
Just wondering how the heck they get 42 elements, are these elements supposed to be relatively prime to 49

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 Post subject:
PostPosted: Sep 29, 2007 - 05:58 PM 
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i guess they take away the elements that can generate the groups ..uhm isn't the order of U(49) is 47 ? Then gcd( 47, 42) = 1

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 Post subject:
PostPosted: Sep 29, 2007 - 06:01 PM 
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I have a theorem saying "If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is phi(d)" Just kind of confused how to use this theorem.

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 Post subject:
PostPosted: Sep 29, 2007 - 08:51 PM 
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U(49) under multiplication modulo 49 ...
Just wondering how the heck they get 42 elements, are these elements supposed to be relatively prime to 49


7, 14, 21, 28, 35, 42, 49 are not relatively prime to 49; if you take away these 7 numbers, you have 42 numbers left. i think this is why U(49) has 42 elements.

but forget about why. they told u that the group is cyclic and has 42 elements. so just use this fact

"If d is a positive divisor of n, the number of elements of order d in a cyclic group of order n is phi(d)"

restate the theorem: Let G be a cyclic group of order n. If d | n, then the number of elements of order d in G is phi(d).

use the theorem: G = U(49) has order n = 42.
we want to know how many generators in G (that is, how many elements of order n; recall that g generates G if G = )
we have n | n. so by theorem, the number of elements of order n in G is phi(n).

so the number of generators in G is phi(n) = phi(42) = ....

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 Post subject: Re: cyclic group
PostPosted: Jan 25, 2018 - 11:11 AM 
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 Post subject: Re: đố ai ?
PostPosted: Apr 05, 2018 - 06:54 PM 
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 Post subject: Re: cyclic group
PostPosted: Apr 25, 2018 - 06:08 PM 
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 Post subject: Re: cyclic group
PostPosted: Apr 25, 2018 - 07:23 PM 
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 Post subject: Re: cyclic group
PostPosted: Apr 27, 2018 - 10:11 PM 
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 Post subject: Re: cyclic group
PostPosted: Jun 27, 2018 - 07:46 PM 
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 Post subject: Re: cyclic group
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PostPosted: Jul 18, 2018 - 12:26 AM 
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 Post subject: Re: cyclic group
PostPosted: Oct 19, 2018 - 06:26 PM 
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~ Tập Yêu ~
~ Tập Yêu ~

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