If G1, G2 are two groups and G = G1 x G2 = {(a,b)|a in G1, b in G2, where we define (a,b)(c,d)=(ac,bd), show that:
a) N={a,e2)|a in G1}, where e2 is the unit element of G2, is a normal subgroup of G.
b) there is an isomorphism from N onto G1.
c) there is an isomorphism from G/N onto G2.
Please show steps. Thank you!