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?