Suppose that (S, ) and (T,*) are isomorphic. Show that (S, ) is commutative if and only if (T,*) is commutative.
Suppose that (S, ) is associative. Is (T,*) associative? Prove your statement.
I am having problems with the above question.
Suppose that (S, ) and (T,*) are isomorphic. Show that (S, ) is commutative if and only if (T,*) is commutative.
Suppose that (S, ) is associative. Is (T,*) associative? Prove your statement.
I am having problems with the above question.
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
Ok this is what i got can u tell me if im right?
If we suppose that (S, ) is assiociative and let x,y,z 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 (b c)) since S is a homomorphsim.
= f((a b) c) since f is commutative.
=f(a b) * f(c) since it is a homomorphism
=(f(a)*f(b))*f(c)
=(x*y)*z