math proof problem help

• Oct 9th 2009, 04:53 PM
koukou8617
math proof problem help
.#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
• Oct 9th 2009, 05:35 PM
aman_cc
Quote:

Originally Posted by koukou8617
#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

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

Are you missing some info on this question. Because for any S with a binary operation * defined, we can have multiple left identities
• Oct 10th 2009, 03:37 AM
Defunkt
Quote:

Originally Posted by aman_cc
Are you missing some info on this question. Because for any S with a binary operation * defined, we can have multiple left identities

I think the question meant to show that there is only one left and right identity element. If that is correct, simply assume that there exist $e_1,e_2 \in S$ such that they are both identity elements. Then,

$e_1e_2 = e_2e_1 = e_1$
but also
$e_1e_2 = e_2e_1 = e_2$

can you finish?
• Oct 10th 2009, 04:30 AM
Swlabr
Quote:

Originally Posted by koukou8617
.#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

Transitivity is the key - essentially, this means that for all $x,y \in S$ there exists a $s_1, s_2 \in S$ such that $x s_1=y$ and $s_2 x=y$ (or certainly that is my understanding of it applied here - essentially you are taking the action of the semigroup over the semigroup and saying it is transitive).

So, if $ex=x$ for some element we have then that $x s_1=y \Rightarrow ey = exs_1 = xs_1 = y$ and so it is a left identity for each element in $S$.

The proof is analogous for the right identity, and then the result follows from Defunkt's post.
• Oct 10th 2009, 08:23 AM
aman_cc
Quote:

Originally Posted by Defunkt
I think the question meant to show that there is only one left and right identity element. If that is correct, simply assume that there exist $e_1,e_2 \in S$ such that they are both identity elements. Then,

$e_1e_2 = e_2e_1 = e_1$
but also
$e_1e_2 = e_2e_1 = e_2$

can you finish?

@Defunkt - Hi-Thanks for your post. I am in doubt if I am following what you wrote. So, let me plz state that and you tell me if I got it correct or not

S - is a set with a binary operation defined
e1 - Right Identity of the set => x.e1 = x, for all x in S
e2 - Left Identity of the set => e2.x = x, for all x in S

Are we trying to show e1=e2=e (unique identity element in S)?
• Oct 10th 2009, 08:33 AM
Defunkt
Quote:

Originally Posted by aman_cc
@Defunkt - Hi-Thanks for your post. I am in doubt if I am following what you wrote. So, let me plz state that and you tell me if I got it correct or not

S - is a set with a binary operation defined
e1 - Right Identity of the set => x.e1 = x, for all x in S
e2 - Left Identity of the set => e2.x = x, for all x in S

Are we trying to show e1=e2=e (unique identity element in S)?

Hi.

What I meant was that both $e_1,e_2$ are left and right identity elements, ie. $\forall x \in S, e_1x = xe_1 = x, e_2x = xe_2 = x$
• Oct 10th 2009, 09:20 AM
Swlabr
Quote:

Originally Posted by Defunkt
Hi.

What I meant was that both $e_1,e_2$ are left and right identity elements, ie. $\forall x \in S, e_1x = xe_1 = x, e_2x = xe_2 = x$

hmm...I didn't read your post fully. Why are you saying you want them both to be identities? You just need to prove that if you have a unique left and a unique right identity then they are equal and you have a unique identity...
• Oct 10th 2009, 09:41 AM
aman_cc
Quote:

Originally Posted by Swlabr
hmm...I didn't read your post fully. Why are you saying you want them both to be identities? You just need to prove that if you have a unique left and a unique right identity then they are equal and you have a unique identity...

Yep - i would agree with you.
• Oct 10th 2009, 09:51 AM
Defunkt
Quote:

Originally Posted by Swlabr
hmm...I didn't read your post fully. Why are you saying you want them both to be identities? You just need to prove that if you have a unique left and a unique right identity then they are equal and you have a unique identity...

You're right. I misread the first post.
• Oct 10th 2009, 01:40 PM
koukou8617
i know how to sole question #1

can you guys tell e how to do this one

Prove that n|φ(a^n-1) for every integer a≥2 and any positive integer n
• Oct 11th 2009, 01:44 AM
Swlabr
Quote:

Originally Posted by koukou8617
i know how to sole question #1

can you guys tell e how to do this one

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

What have you done on this problem so far?
• Oct 11th 2009, 08:14 AM
koukou8617
Quote:

Originally Posted by Swlabr
What have you done on this problem so far?