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Generator of the additive gr. Z_m 
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#1
Nov2008, 12:32 AM

P: 1,635

Hi all,
I am having trouble proving the following proposition: [tex]\bar a[/tex] is a generator of the additive group [tex]Z_m [/tex] if and only if gcm(m,a)=1. Well, first let's start with what i know. I know how to prove the following: Let G=[a] be a cyclic group of order q. Show that [tex] a^s[/tex] is a generator of G, iff gcd(s,q)=1. proof: (<=) Suppose that gcd(s,q)=1, then there exist some integers x,y such that 1=sx+qy, so [tex]a^1=a^{sx}a^{qy}=(a^s)^x(a^q)^y=(a^s)^xe^y=(a^s)^x[/tex] so since there exists an integer x, such that [tex] (a^s)^x=a[/tex] then [tex] G=[a^s][/tex] (=>) Now, suppose that [tex] a^s[/tex] is a generator of G. THen there should exist some integer k such that [tex] (a^s)^k=a=>a^{sk}=a=>a^{sk1}=e[/tex] now from a theorem we know that q(sk1)=> there exists some integer n such that sk1=mq=>1=sk+(m)q=>gcd(s,q)=1. Now i tried to translate this for the additive groups. And here is where the problems start to come in play, for only to get worse when i go to Z_m. So, i am trying to prove the following: Let G be an additive group generated by a, where o(G)=q. Prove that s*a is a generator of G iff gcd(s,q)=1. Proof: (<=) Suppose that gcd(s,q)=1. Now, as before, there exist x,y integers, such that 1=sx+qy now: 1*a=(sx+qy)a=(sx)a+(qy)a= x(sa)+y(qa).(@) (Now, here i believe that if we Translate the Lagranges theorem into terms of an additive group it would be sth like this, right: "Let G be a finite group with order r. Then the order of each subgroup H in G, and the order of each element a of G is an integral divisor or r. Also r*g=0 for every el. g in G."Basically i am concerned whether the last part would be correct that is: from g^r =e into r*g=e=0. Since in the book we are using we are denoting with 0 the identity in an additive group.) Now, if this is true, then we get from (@) a=x(sa). Now since there is an integer x, such that this holds, i assume we can conclude that G=[sa], that is sa generates the group G. (=>) now lets suppose that sa is a generator for the group G. Then there exists some integer k such that k(sa)=a=> k(sa)a=0 => (ks1)a=0. Now, since the order of a is q. it follows that q(ks1)=> ks1=mq =>1=ks +(m)q , so it follows that gcd(s,q)=1. Well, let me give a crack to my main issue now: [tex]\bar a[/tex] is a generator of the additive group [tex]Z_m [/tex] if and only if gcm(m,a)=1. Proof: again lets suppose that gcd(a,m)=1. so there are integers x, y such that 1=ax+my. Now, i know that [tex]\bar a = a+[m][/tex] so let [tex][\bar a]=\{k\bar a:k\in Z\}=\{ka+k(rm):r,k\in Z\}[/tex] in particular let [tex] u\in [\bar a][/tex] so, [tex]u=ak+k(rm)[/tex] So, this would mean that any linear combination of a and m is also in a+[m]. Now since 1=ax+my, i am saying that [tex]1\in a+[m]=\bar a[/tex] ok let's stop here, cuz, i lost my stream of thought! Any hints, ideas, would be greatly appreciated. 


#2
Nov2008, 12:55 AM

P: 1,635

Well, i know the previous post is way too long, so if you don't have time to have a look at it, the whole thing is about the following:
Prove that [tex]\bar a[/tex] is a generator of the additive group [tex]Z_m[/tex] if and only if gcm(m,a)=1. 


#3
Nov2008, 07:51 PM

P: 1,635

Nevermind, i figuret it out. Thnx though!



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