(Z/pZ)* isomoprhic to Z/(p-1)Z?

jinydu · 2008-06-02, 05:52

In proving a much larger theorem, my abstract algebra textbook assumes this result without proof. Here p is a prime, (Z/pZ)* is the group of units of Z/pZ, i.e. {1, 2, ... p-1} with multiplication modulo p as the operation and Z/(p-1)Z is {0, 1, ... p-2} with addition modulo p-1 as the operation.

This claim is not at all obvious to me. How do I justify it?

Thanks

akruppa · 2008-06-02, 06:57

Both are cyclic, and they have the same order.

jinydu · 2008-06-02, 07:50

Thanks. The hint from my teaching assistant was that the group of units of any finite field is cyclic. Unfortunately, I don't know the proof of that and I've been trying to find it. I did find one proof online, but it has a hole in it. Here's my paraphrasing of it, applied to the specific case I'm interested in:

We can regard (Z/pZ)* as a subset of the field Z/pZ. Since |(Z/pZ)*| = p-1, the order of any element in (Z/pZ)* divides p-1. Also, for any divisor d of p-1, the number of elements such that x^d = 1 is at most d because t^d - 1 is a polynomial in Z/pZ and hence has at most d solutions. Let D be the least common multiple of the orders of all the elements in (Z/pZ)*. Then every element is a root of the polynomial t^D - 1. Since there are only p-1 elements, this forces D >= p-1. Since p-1 is a multiple of all the orders, D <= p-1. Thus D = p-1. ... Choose an element with order D... The existence of an element of order p-1 shows that (Z/pZ) is cyclic.

The problem here is the existence of an element whose order is the least common multiple of all the orders. It's not clear to me why such an element must exist.

I do know that in abelian group, order(xy) = order(x)order(y) whenever order(x) and order(y) are relatively prime. But once I drop the condition on relative primality, the result fails completely; I can't even claim that the right hand side is the least common multiple of order(x) and order(y).

Peter Hackman · 2008-09-24, 10:22

I suppose it's too late for your purposes, but then I'm not breaking
the rules.

I'm surprised no one has answered your question. It's a standard result
in Elementary Number Theory and its proof is given in just about any book on that topic, as well as in several more advanced texts.

There are proofs on pp. 70 and 177 in the following document:

http://www.mai.liu.se/~pehac/booktot.pdf

ET_ · 2008-09-24, 10:32

Quote:

Originally Posted by Peter Hackman

I suppose it's too late for your purposes, but then I'm not breaking
the rules.

I'm surprised no one has answered your question. It's a standard result
in Elementary Number Theory and its proof is given in just about any book on that topic, as well as in several more advanced texts.

There are proofs on pp. 70 and 177 in the following document:

http://www.mai.liu.se/~pehac/booktot.pdf

Code:

Fel 404: Den sida som du försökte nå är odefinierad,
i likhet med uttrycket 0/0.

Luigi

Peter Hackman · 2008-09-24, 10:51

Quote:

Originally Posted by ET_

Code:

Fel 404: Den sida som du försökte nå är odefinierad,
i likhet med uttrycket 0/0.

Luigi

Sorry:

http://www.mai.liu.se/~pehac/kurser/TATM54/booktot.pdf

jinydu · 2008-09-24, 11:34

Quote:

Originally Posted by Peter Hackman

I suppose it's too late for your purposes, but then I'm not breaking
the rules.

Far too late. Not only have I finished the course, I've already graduated from university. Anyway, I did eventually finish the problem in time.

ATH · 2008-09-24, 13:53

Quote:

Originally Posted by ET_

Code:

Fel 404: Den sida som du försökte nå är odefinierad,
i likhet med uttrycket 0/0.

Luigi

Translation: "The page that you tried to reach is undefined, like the expression 0/0."

ET_ · 2008-09-24, 14:06

Quote:

Originally Posted by ATH

Translation: "The page that you tried to reach is undefined, like the expression 0/0."

Thank you. I had some idea thanks to that 404 and my little knowledge of German

Luigi

2008-06-02, 05:52	#1
jinydu Dec 2003 Hopefully Near M48 2×3×293 Posts	*(Z/pZ) isomoprhic to Z/(p-1)Z?** In proving a much larger theorem, my abstract algebra textbook assumes this result without proof. Here p is a prime, (Z/pZ)* is the group of units of Z/pZ, i.e. {1, 2, ... p-1} with multiplication modulo p as the operation and Z/(p-1)Z is {0, 1, ... p-2} with addition modulo p-1 as the operation. This claim is not at all obvious to me. How do I justify it? Thanks

2008-06-02, 07:50	#3
jinydu Dec 2003 Hopefully Near M48 2×3×293 Posts	Thanks. The hint from my teaching assistant was that the group of units of any finite field is cyclic. Unfortunately, I don't know the proof of that and I've been trying to find it. I did find one proof online, but it has a hole in it. Here's my paraphrasing of it, applied to the specific case I'm interested in: We can regard (Z/pZ)* as a subset of the field Z/pZ. Since \|(Z/pZ)\| = p-1, the order of any element in (Z/pZ) divides p-1. Also, for any divisor d of p-1, the number of elements such that x^d = 1 is at most d because t^d - 1 is a polynomial in Z/pZ and hence has at most d solutions. Let D be the least common multiple of the orders of all the elements in (Z/pZ). Then every element is a root of the polynomial t^D - 1. Since there are only p-1 elements, this forces D >= p-1. Since p-1 is a multiple of all the orders, D <= p-1. Thus D = p-1. ... Choose an element with order D... The existence of an element of order p-1 shows that (Z/pZ) is cyclic. The problem here is the existence of an element whose order is the least common multiple of all the orders. It's not clear to me why such an element must exist. I do know that in abelian group, order(xy) = order(x)order(y) whenever order(x) and order(y) are relatively prime. But once I drop the condition on relative primality, the result fails completely; I can't even claim that the right hand side is the least common multiple of order(x) and order(y). Last fiddled with by jinydu on 2008-06-02 at 07:52*

2008-09-24, 10:22	#4
Peter Hackman Oct 2007 linköping, sweden 2²×5 Posts	I suppose it's too late for your purposes, but then I'm not breaking the rules. I'm surprised no one has answered your question. It's a standard result in Elementary Number Theory and its proof is given in just about any book on that topic, as well as in several more advanced texts. There are proofs on pp. 70 and 177 in the following document: http://www.mai.liu.se/~pehac/booktot.pdf Last fiddled with by Peter Hackman on 2008-09-24 at 10:23

2008-06-02, 06:57	#2
akruppa "Nancy" Aug 2002 Alexandria 4643₈ Posts	Both are cyclic, and they have the same order.