# The question in the name of Normal Subgroup and Isomorphism

Let **$A$** and **$B$** be two groups.
Show that set **$N$ = { $(a,1): a \in A $ }** is the normal subgroup of **$A$ x $B$** and

that the **$A$ x $B$ $/ N$** quotient group is isomorph to **$B$**.

if you help me, i'll be exulted.

Homework ?

yes i can observe that

N=ker(π)

where π:A×B→B is the Projection map of the second factor B.

The kernel of an homomorphism of groups is always normal, so N is a normal subgroup.

Moreover, by the First isomorphism theorem, it holds

(A×B)/ker(π)≅B ... but is it enough?