Cyclic group Generators

3rd year Pure maths · #1 $Old$ November 15th, 2009, 07:52 AM

I'm currently writing up a project on Congruences, Units and Exponents of congruence classes and such like... I have found a quote on wikipedia which references a book called Contemporary abstract algebra by Joseph Gallian. The quote is

'If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler phi function'

Now I can prove this if need be but we are supposed to reference things and Wikipedia is not referenceable. I was wonderring if anyone could confirm whether the quote is in the book or correct me on it. The book isnt in our university library and the similiar books that were dont seem that good IMO

tonio · #2 $Old$ November 15th, 2009, 08:52 AM

Quote:

Originally Posted by 3rd year Pure maths $View Post$

I'm currently writing up a project on Congruences, Units and Exponents of congruence classes and such like... I have found a quote on wikipedia which references a book called Contemporary abstract algebra by Joseph Gallian. The quote is

'If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler phi function'

Now I can prove this if need be but we are supposed to reference things and Wikipedia is not referenceable. I was wonderring if anyone could confirm whether the quote is in the book or correct me on it. The book isnt in our university library and the similiar books that were dont seem that good IMO

I couldn't find such a quote in Gallian's book in the chapter about cyclic groups, and the closest thing is " $\mbox {An integer k is a generator of } \mathbb{Z}_n \mbox{ iff gcd(k,n)=1}$ " , in page 66, corollary to theorem 4.2 (2nd. edition).
This, together with the definition of Euler's Totient Function $\phi$ (page 71, after example 5) gives us at once what you wrote...but there's no such quote in the book, at least in this part of it. Quotes of books without the pages and/or chapters and sections may be something very hard to find sometimes.

Tonio

Amer · #3 $Old$ November 15th, 2009, 09:04 AM

Quote:

Originally Posted by 3rd year Pure maths $View Post$

I'm currently writing up a project on Congruences, Units and Exponents of congruence classes and such like... I have found a quote on wikipedia which references a book called Contemporary abstract algebra by Joseph Gallian. The quote is

'If n is finite, then there are exactly φ(n) elements that generate the group on their own, where φ is the Euler phi function'

Now I can prove this if need be but we are supposed to reference things and Wikipedia is not referenceable. I was wonderring if anyone could confirm whether the quote is in the book or correct me on it. The book isnt in our university library and the similiar books that were dont seem that good IMO

I do not know if you have this book
book name : A first course in Abstract Algebra ,third addition
John B.Fraleigh , Department of Mathematics University of Rhode Island
ADDISON-WESLEY
PUBLISHING COMPANY
Reading, Massachusetts, Amsterdam , London , Manila, Singapore , Sydney, Tokyo

Page 62
there is a corollary after the theorem 6.3 that said
"""
if "a" is a generator of a finite cyclic group G of order n, then the other generators of G are the elements of the form $a^r$ , where r is relatively prime to n, that is, where the greatest common divisor of r and n is 1.
"""

3rd year Pure maths · #4 $Old$ November 15th, 2009, 12:33 PM

Yeh I have taken that book out of the Library and in addition have a book by I N Herstein... both books arent bad but theyre not really as good as they could be.

I can prove everything I need to if it comes to it but this module we get as many (if not more marks) for demonstrating our ability to research, reference and communicate professionally as we do for the actual maths.

I think I will demonstrate the proof I have and reference Gallian as was just listed. I already have a neat proof for Eulers formula