Math proof problem help

1. Oct 9, 2009

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

Last edited: Oct 10, 2009
2. Oct 9, 2009

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?

3. Oct 10, 2009

koukou

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

4. Oct 10, 2009

honestrosewater

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

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

Last edited: Oct 10, 2009
5. Oct 10, 2009

koukou

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

6. Oct 11, 2009

honestrosewater

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

7. Oct 12, 2009

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.