Could somebody answer this question Thank you A Examples of

Could somebody answer this question?

Thank you

A. Examples of the FHT Applied to Finite Groups In each of the following. use the fundamental homomorphism theorem to prove that the two given groups are isomorphic. Then display their tables. Example Z2 and Z/(2). f =(0 1 2 3 4 5 0 1 0 1 0 1 is a homomorphism from Z6 onto Z2. (Do not prove that f is a homomorphism.) The kernel off is (0,2,4) (2). Thus, It follows by the FHT that Z2 partially equals to Z6/(2). 3 Z2 and S3/{epsilon, beta,sigma}.

Solution

The kernel of f is the elements which are mapped onto 0

K = (0,2,4)

Consider the mapping z6 to z2

Thus f(x) = 0 if x is odd and f(x) = even if x is even.

Consider f(x*y) if both x and y are odd

f(x*y) = f(odd) = 1 = f(x)*f(y)

If any one of x or y is even, then

f(x*y) = f( even) =0 = f(x)*f(y)

Hence it follows tha Z6 is isomorphic to z2

By fundamental homo theorem hence it is proved that Z2 is homomorphic to Z6/(2)