1. ## Commutative Isomorphic?

Suppose that (S, $\displaystyle \bullet$) and (T,*) are isomorphic. Show that (S, $\displaystyle \bullet$) is commutative if and only if (T,*) is commutative.

Suppose that (S, $\displaystyle \bullet$) is associative. Is (T,*) associative? Prove your statement.

I am having problems with the above question.

2. Let f be an isomorphism between S and T

let s,t\in T . Given that S is commutative, we want to show that st=ts.
Since f is isomorphism, f is onto. So there are a,b such that f(a)=s and f(b)=t

Now

st=f(a)f(b)
=f(a\bullet b) by definition of homomorphism
=f(b\bullet a) since S is commutative
=f(b)f(a) again by homomorphism
=ts

For associativity you may try by yourself first

3. Ok this is what i got can u tell me if im right?

If we suppose that (S,$\displaystyle \bullet$) is assiociative and let x,y,z$\displaystyle \in$T. Since f is isomorphic f is onto, so there are a,b,c such that f(a)=x, f(b)=y, f(c)=z

x*(y*z)=f(a $\displaystyle \bullet$ (b$\displaystyle \bullet$ c)) since S is a homomorphsim.
= f((a $\displaystyle \bullet$ b) $\displaystyle \bullet$ c) since f is commutative.
=f(a $\displaystyle \bullet$ b) * f(c) since it is a homomorphism
=(f(a)*f(b))*f(c)
=(x*y)*z

4. Originally Posted by I Congruent
Ok this is what i got can u tell me if im right?

If we suppose that (S,$\displaystyle \bullet$) is assiociative and let x,y,z$\displaystyle \in$T. Since f is isomorphic f is onto, so there are a,b,c such that f(a)=x, f(b)=y, f(c)=z

x*(y*z)=f(a $\displaystyle \bullet$ (b$\displaystyle \bullet$ c)) since S is a homomorphsim.
= f((a $\displaystyle \bullet$ b) $\displaystyle \bullet$ c) since f is commutative.
=f(a $\displaystyle \bullet$ b) * f(c) since it is a homomorphism
=(f(a)*f(b))*f(c)
=(x*y)*z
Becareful, don't skip steps
Let me do first
$\displaystyle x*(y*z)=f(a)*(f(b)*f(c) )$ just relacing x with f(a) and so on
$\displaystyle =f(a)*(f(b\bullet c))$ definition of homomorphism

Now try to continue
Edit: well you don't need to because that two steps above is exactly the steps that are missing

5. Thank you very much i think i finally understand this.