The question in the name of Normal Subgroup and Isomorphism

asked 2020-01-01 16:04:29 +0100

updated 2020-06-01 10:50:04 +0100

FrédéricC gravatar image

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.

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Homework ?

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2020-01-01 23:15:03 +0100 )edit

yes i can observe that


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?

hayyambey gravatar imagehayyambey ( 2020-01-02 17:56:39 +0100 )edit