proof question? please help?

Question

let A and B be groups. Prove that A X B is isomorphic to
B X A .

Answer 1

Let A group have c and d and let B group have e and f.
A X B = B X A means that
(c+d) (e+f) = (e+f) (c+d)
ce+cf+de+df = ec+ed+fc+fd

Now let c=2, d=3, e=4, f=5
2*4+2*5+3*4+3*5 = 4*2+4*3+5*2+5*3
8+10+12+15 = 8+12+10+15
45=45

Thus A X B = B X A is correct. and of course isomorphic too.

Answer 2

consider the following map
F: A X B ---> B X A
F(a,b)=(b,a)

it is a homomorphism,
its kernel is (e_A,e_B)
and it is surjective
therefore it is an isomorphism