What is the proof for n|φ(a^n-1)?

[koukou]

#1 a) If ex = x for some elements e,x belong to S, we say e is a left identity for x; similarly, if xe = x we say e is a right identity for x. Prove that an element is a left identity for one element of S if and only if it is a left identity for every element of S. Let S be a non-empty set with a binary operation which is associative and both left and right transitive

b) Prove that S has a unique identity element

c) Deduce that S is a group under the given binary operation



#2.Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n

 

[honestrosewater]

How do you start? For (a), the second implication is immediate. Do you have other conditions for the operation? Is it associative? Do you have inverses? Is S the domain of a group?

 

[koukou]

How do you start? For (a), the second implication is immediate. Do you have other conditions for the operation? Is it associative? Do you have inverses? Is S the domain of a group?

there exists a such that and xs1=y s2x=y..

 

[honestrosewater]

there exists a such that and xs1=y s2x=y..

Part of your post got lost. Is this an axiom?

I missed your edit. What does it mean to be left and right transitive?

 

[koukou]

Part of your post got lost. Is this an axiom?

I missed your edit. What does it mean to be left and right transitive?

thank you
i have done this one

but still no idea to do this one


Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n

 

[honestrosewater]

Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n

What does that formula say? What do "|", "φ", and "φ(x)" mean?

 

[MathematicalPhysicist]

For 2, use the fact that:
φ(n)=(p1^k1-p1^(k1-1))...(pr^kr-pr^kr+1)
for n= p1^k1 ... pr^kr
for pi primes, and because a^n-1=(a-1)(a^n-1+...+1)
Now prove this theorem by induction.